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Polynomial and
Rational Functions
Polynomial Functions and Models
Graphing Polynomial Functions
Polynomial Division; The Remainder and Factor Theorems
Zeroes of Polynomial Functions and Their Theorems
Rational Functions
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Polynomial and
Rational Functions
(continued)
Inequalities
Variation and Problem-Solving
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Polynomial and Rational Functions > Polynomial Functions and Models
Polynomial Functions and Models
• The Leading-Term Test
• Finding Zeroes of Factored Polynomials
• Introduction: Polynomial and Rational Functions and Models
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Polynomial and Rational Functions > Polynomial Functions and Models
The Leading-Term Test
• Properties of the leading term of a polynomial reveal whether the function
increases or decreases continually as x values approach positive and negative
infinity.
• If n is odd and an is positive, the function declines to the left and inclines to the
right.
• If n is odd and an is negative, the function inclines to the left and declines to the
right.
• If n is even and an is positive, the function inclines both to the left and to the right.
• If n is even and an is negative, the function declines both to the left and to the
right.
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Polynomial and Rational Functions > Polynomial Functions and Models
Finding Zeroes of Factored Polynomials
• A polynomial function may have zero, one, or many zeros.
• All polynomial functions of positive, odd order have at least one zero, while
polynomial functions of positive, even order may not have a zero.
• Regardless of odd or even, any polynomial of positive order can have a maximum
number of zeros equal to its order.
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Polynomial and Rational Functions > Polynomial Functions and Models
Introduction: Polynomial and Rational Functions and Models
• Researchers will often collect many discrete samples of data, relating two or more
variables, without knowing the mathematical relationship between them. Curve
fitting is used to create trend lines intended to fill in the points between and
beyond collected data points.
• Polynomial functions are easy to use for modeling but ill-suited to modeling
asymptotes and some functional forms, and they can become very inaccurate
outside the bounds of the collected data.
• Rational functions can take on a much greater range of shapes and are more
accurate both inside and outside the limits of collected data than polynomial
functions. However, rational functions are more difficult to use and can include
Curve Fitting
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undesirable asymptotes.
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Polynomial and Rational Functions > Graphing Polynomial Functions
Graphing Polynomial Functions
• Basics of Graphing Polynomial Functions
• The Intermediate Value Theorem
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Polynomial and Rational Functions > Graphing Polynomial Functions
Basics of Graphing Polynomial Functions
• The graph of the zero polynomial f(x) = 0 is the x-axis.
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Polynomial and Rational Functions > Graphing Polynomial Functions
• The graph of a degree 1 polynomial (or linear function) [Equation 1], where a1 ≠ 0, is an oblique line
with y-intercept a0 and slope a1.
Equation 1
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Polynomial and Rational Functions > Graphing Polynomial Functions
• The graph of a degree 2 polynomial [Equation 2], where a2 ≠ 0 is a parabola.
Equation 2
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Polynomial and Rational Functions > Graphing Polynomial Functions
• The graph of any polynomial with degree 2 or greater [Equation 3], where an ≠ 0 and n ≥ 2 is a
continuous non-linear curve.
Equation 3
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Polynomial and Rational Functions > Graphing Polynomial Functions
The Intermediate Value Theorem
• Simply stated, the Intermediate Value Theorem points out that: if the plotted route
between points A and C is smooth and continuous between point A to point C,
you will have to pass through all points "B" on the journey, as long as they are on
the plotted route.
• The Intermediate Value Theorem capitalizes on the completeness of functions of
real numbers.
• Functions containing irrational roots do not meet the requirements of the
Intermediate Value Theorem.
The Intermediate Value Theorem
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Polynomial and Rational Functions > Polynomial Division; The Remainder and Factor T...
Polynomial Division; The Remainder and Factor Theorems
• Division and Factors
• The Remainder Theorem and Synthetic Division
• Finding Factors of Polynomials
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Polynomial and Rational Functions > Polynomial Division; The Remainder and Factor T...
Division and Factors
• Dividing one polynomial by another can be achieved by using long division. The
rules for polynomial long division are the same as the rules learned for long
division of integers.
• The four steps of long division are divide, multiply, subtract, and bring down.
• After completing polynomial long division, it is good to check the answers, either
by plugging in a number or by multiplying the quotient times the divisor to get the
dividend back.
745 divided by 3
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Polynomial and Rational Functions > Polynomial Division; The Remainder and Factor T...
The Remainder Theorem and Synthetic Division
• Synthetic division is most commonly applied when dividing by a monomial such
as x-a.
• The most useful aspects of synthetic division are that it allows one to calculate
without writing variables and uses fewer calculations.
• In algebra, synthetic division is a method of performing polynomial long division,
with less writing and fewer calculations.
The steps of synthetic division
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Polynomial and Rational Functions > Polynomial Division; The Remainder and Factor T...
Finding Factors of Polynomials
• Factoring is a critical skill in simplifying functions and solving equations.
• There are four types of factoring shown which are "pulling out" common factors,
factoring perfect squares, the difference between two squares, and then how to
factor when the other three techniques are not applicable.
• The first step should always be "pulling out" common factors. Even if this does not
factor out the polynomial completely, this will make the rest of the process much
easier.
FOIL Method Diagram
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Zeroes of Polynomial Functions and Their Theorems
• The Fundamental Theorem of Algebra
• Finding Polynomials with Given Zeroes
• Zeroes of Polynomial Functions with Real Coefficients
• Rational Coefficients
• Integer Coefficients and the Rational Zeroes Theorem
• The Rule of Signs
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
The Fundamental Theorem of Algebra
• The fundamental theorem of algebra states that every non-constant singlevariable polynomial with complex coefficients has at least one complex root. This
includes polynomials with real coefficients, since every real number is a complex
number with zero imaginary part.
• Equivalently (by definition), the fundamental theorem states that the field of
complex numbers is algebraically closed.
• The fundamental theorem is also stated as follows: every non-zero, singlevariable, degree n polynomial with complex coefficients has, counted with
multiplicity, exactly n roots. The equivalence of the two statements can be proven
through the use of successive polynomial division.
The Maximum Modulus Principle
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Finding Polynomials with Given Zeroes
• A polynomial constructed from n roots will have degree n or less. That is to say, if
given three roots, then the highest exponential term needed will be x3.
• Each zero given will end up being one term of the factored polynomial. After
finding all the factored terms, simply multiply them together to obtain the whole
polynomial.
• Because a polynomial and a polynomial multiplied by a constant have the came
roots, every a polynomial is constructed from given zeroes the general solution
includes a constant, shown here as c.
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Zeroes of Polynomial Functions with Real Coefficients
• Real numbers include all the rational and irrational numbers. For example: −5
,4/3, √2 are all real numbers.
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
• If given the function, [Equation 4], any value of [Equation 5] that will result in [Equation 6] is a root of
the function. For this reason, roots are often referred to as a zero of the function.
Equation 4
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Equation 5
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Equation 6
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
• There are many ways to find the roots of a polynomial. If one is confident factoring out polynomials into their simplest forms, its
roots can usually be found by inspection. However, if one is not confident, or it is a tricky polynomial, the quadratic equation
can be used.
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Rational Coefficients
• In mathematics, a rational number is any number that can be expressed as the
quotient or fraction p/q of two integers, with the denominator q not equal to zero.
• A real number that is not rational is called irrational. Irrational numbers include √2,
π, and e.
• Polynomials with rational coefficients can be treated just like any other
polynomial, just remember to utilize all the properties of fractions necessary
during your operations.
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Integer Coefficients and the Rational Zeroes Theorem
• In algebra, the Rational Zeros Theorem (also known as Rational Root Theorem, or Rational Root
Test) states a constraint on rational solutions (or roots) of the polynomial equation [Equation 7]with
integer coefficients.
Equation 7
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
Integer Coefficients and the Rational Zeroes Theorem
• If a0 and an are non-zero, then each rational solution x, when written as a fraction
x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies
1) p is an integer factor of the constant term a0, and 2) q is an integer factor of the
leading coefficient an.
• A proof can be derived by first moving the constants to one side, factoring and
multiplying by qn. Then a generalized form of Euclid's lemma states that p divides
a0. The proof for q is similar.
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Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems
The Rule of Signs
• The rule of signs gives us an upper bound number of positive or negative roots of
a polynomial. It is not a complete criterion, meaning that it does not tell the exact
number of positive or negative roots.
• The rule states that if the terms of a polynomial with real coefficients are ordered
by descending variable exponent, then the number of positive roots of the
polynomial is either equal to the number of sign differences between consecutive
nonzero coefficients, or is less by a multiple of 2.
• As a corollary of the rule, the number of negative roots is the number of sign
changes after multiplying the coefficients of odd-power terms by −1 [f(-x)], or
fewer than it by a multiple of 2.
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Polynomial and Rational Functions > Rational Functions
Rational Functions
• Finding the Domain of a Rational Function
• Asymptotes
• Solving Problems with Rational Functions
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Polynomial and Rational Functions > Rational Functions
Finding the Domain of a Rational Function
• A rational function is any function which can be written as the ratio of two
polynomial functions.
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Polynomial and Rational Functions > Rational Functions
• The domain of [Equation 8] is the set of all points x for which the denominator Q(x) is not zero,
where one assumes that the fraction is written in its lower degree terms, that is, P and Q have several
factors of the positive degree.
Equation 8
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Polynomial and Rational Functions > Rational Functions
• Domain restrictions can be determined by setting the denominator equal to zero and solving.
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Polynomial and Rational Functions > Rational Functions
Asymptotes
• An asymptote of a curve is a line such that the distance between the curve and
the line approaches zero as they tend to infinity.
• There are potentially three kinds of asymptotes: horizontal, vertical and oblique
asymptotes.
• A rational function has at most one horizontal asymptote or oblique (slant)
asymptote, and possibly many vertical asymptotes.
Asymptotes
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Polynomial and Rational Functions > Rational Functions
Solving Problems with Rational Functions
• A rational function is defined as the ratio of two real polynomials with the condition
that the polynomial in the denominator is not a zero polynomial.
• The x-intercepts, also known as zeros of the function or real roots, can be more
than one x-intercept. On graphs, x-intercepts are points where a graph intersects
the x-axis. Thus, x-intercepts are x-values for which the function has a value of
zero.
• In the case of rational functions, the x-intercepts exist when the numerator is
equal to 0. In the case of rational functions, the x-intercepts exist when the
numerator is equal to 0. For f(x) = P(x)/Q(x), if P(x) = 0, then f(x) = 0.
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Polynomial and Rational Functions > Inequalities
Inequalities
• Polynomial Inequalities
• Rational Inequalities
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Polynomial and Rational Functions > Inequalities
Polynomial Inequalities
• To solve a polynomial inequality, first rewrite the polynomial in factored form to
find its zeros.
• For each zero, input the value of the zero in place of x in the polynomial.
Determine the sign (positive or negative) of the polynomial as it passes the zero in
the rightward direction.
• Determine the intervals between these roots which satisfy the inequality.
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Polynomial and Rational Functions > Inequalities
Rational Inequalities
• First factor the numerator and denominator polynomial to reveal the zeros in
each.
• Substitute x with a zero (root) to determine whether the rational function is
positive or negative to the right of that point. Repeat for all zeros.
• The intervals that satisfy the inequality symbol will be the answer. Note that for
any ≥ or ≤, the interval will only be closed to include the zero if the zero is found in
the numerator. If the zero is found in the denominator, that point is undefined, and
cannot be included in the solution.
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Polynomial and Rational Functions > Variation and Problem-Solving
Variation and Problem-Solving
• Direct Variation
• Inverse Variation
• Combined Variation
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Polynomial and Rational Functions > Variation and Problem-Solving
Direct Variation
• The ratio of variables in direct variation is always constant
• Direct variation between variables is easily modeled using a linear graph.
• The equation relating directly varying variables to a constant can be rearranged to
slope-intercept form.
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Polynomial and Rational Functions > Variation and Problem-Solving
Inverse Variation
• The ratio of variables in direct variation is always constant.
• Direct variation between variables is depicted by an hyperbola.
• The equation relating indirectly varying variables to a constant can be rearranged
to hyberbolic form.
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Polynomial and Rational Functions > Variation and Problem-Solving
Combined Variation
• There must be a minimum of three related variables for their relationship to be
one of combined variation.
• Among the three or more related variables, one must directly vary with another
and inversely vary with a third in order for the relationship to be one of combined
variation.
• An example of combined variation in the physical world is the Combined Gas
Law, which relates pressure, temperature, volume, and moles (amount of
molecules) of a gas.
Illustration of Gay-Lussac's Law, derived from
the Combined Gas Law
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Appendix
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Polynomial and Rational Functions
Key terms
• asymptote A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity".
• asymptote A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity".
• common factor A value, variable or combination of the two that is common to all terms of a polynomial.
• constant An identifier that is bound to an invariant value.
• constant An identifier that is bound to an invariant value.
• constant An identifier that is bound to an invariant value.
• continuous Without break, cessation, or interruption; without intervening time.
• coprime Having no positive integer factors, aside from 1, in common with one or more specified other positive integers.
• denominator The number or expression written below the line in a fraction (thus 2 in ½).
• denominator The number or expression written below the line in a fraction (thus 2 in ½).
• directly proportional If one variable is always the product of the other and a constant, the two are said to be directly
proportional.
• dividend A number or expression that is to be divided by another.
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Polynomial and Rational Functions
• divisor An integer that divides another integer an integral number of times.
• domain The set of all points over which a function is defined.
• Euclid's lemma In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures one of the
fundamental properties of prime numbers. It states that if a prime divides the product of two numbers, it must divide at least one
of the factors. For example since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7
= 133. It is used in the proof of the fundamental theorem of arithmetic.
• factor To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).
• hyperbola A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not
tangent to the cone.
• indeterminate not accurately determined or determinable.
• inequality A statement that of two quantities one is specifically less than or greater than another. Symbols: < or ≤ or > or ≥, as
appropriate.
• inequality A statement that of two quantities one is specifically less than or greater than another. Symbols: < or ≤ or > or ≥, as
appropriate.
• interval A distance in space.
• irrational number Any real number that cannot be expressed as a ratio of two integers.
• Leading coefficient The coefficient of the leading term.
• Leading term The term in a polynomial in which the independent variable is raised to the highest power.
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Polynomial and Rational Functions
• multiplicity the number of values for which a given condition holds
• numerator The number or expression written above the line in a fraction (thus 1 in ½).
• oblique Not erect or perpendicular; neither parallel to, nor at right angles from, the base; slanting; inclined.
• polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient
and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0 .
Importantly, because all exponents are positive, it is impossible to divide by x.
• polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient
and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0.
Importantly, because all exponents are positive, it is impossible to divide by x.
• polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient
and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0.
Importantly, because all exponents are positive, it is impossible to divide by x.
• proportional At a constant ratio (to). Two magnitudes (numbers) are said to be proportional if the second varies in a direct
relation arithmetically to the first.
• quotient The number resulting from the division of one number or expression by another.
• quotient The number resulting from the division of one number or expression by another.
• rational function Any function whose value can be expressed as the quotient of two polynomials (except division by zero).
• rational function Any function whose value can be expressed as the quotient of two polynomials (except division by zero).
• rational function Any function whose value can be expressed as the quotient of two polynomials (except division by zero).
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• real number An element of the set of real numbers. The set of real numbers include the rational numbers and the irrational
numbers, but not all complex numbers.
• remainder The amount left over after subtracting the divisor as many times as possible from the dividend without producing a
negative result. If (dividend) and d (divisor) are integers, then can always be expressed in the form n = dq + r, where q
(quotient) and r (remainder) are also integers and 0 ≤ r < d.
• root the number which,when plugged into the equation, will produce a zero.
• root the number which,when plugged into the equation, will produce a zero.
• sign positive or negative polarity.
• term any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an
overall expression or table.
• y-intercept A point at which a line crosses the y-axis of a Cartesian grid.
• zero Also known as a root, a zero is an x value at which the function of x is equal to 0.
• zero Also known as a root, a zero is an x value at which the function of x is equal to 0.
• zero Also known as a root, a zero is an x value at which the function of x is equal to 0.
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Polynomial and Rational Functions
The Intermediate Value Theorem
In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the
Intermediate Value Theorem.
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Polynomial and Rational Functions
Interactive Graph: Graphing a Rational Function
A graph of a rational function, . A discontinuity occurs when : the function is not defined at $x=-2$.
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Polynomial and Rational Functions
Interactive Graph: Continuous Function
A graphed third-order equation where . It meets the requirements of the Intermediate Value Theorem. In what situation would it not meet the
requirements for the theorem?
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Interactive Graph: Cubic Function in Factored Form
Graph of cubic function in factored form of . Notice that f(x) crosses the x axis at x=-3, x=-1, and x=2.
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Polynomial and Rational Functions
Interactive Graph: Graph of Fourth-Degree Polynomial
Graph of the fourth-degree polynomial with the equation . This polynomial has four roots. It is positive in three segments and negative in two. If it were a
polynomial inequality with the condition that all values are greater than 0, the two negative segments would be removed.
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Polynomial and Rational Functions
Interactive Graph: Direct Variation
Graph of direct variation with the linear equation y=0.8x. The line y=kx is an example of direct variation between variables x and y. For all points on the
line, y/x=k. Notice what happens when you change the "k" term.
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Polynomial and Rational Functions
Interactive Graph: Graph of a Cubic Polynomial
Graph of a cubic polynomial with the quadratic equation . How does the shape change when you change the a term? When you change the n term?
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Polynomial and Rational Functions
Interactive Graph: Graph of 3rd-Degree Polynomial
Graph of a polynomial function of a degree 3, with the equation . Notice how many times the graph crosses the x-axis.
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Polynomial and Rational Functions
Interactive Graph: Graph of 4th-Degree Polynomial
Graph of a polynomial function of a degree 4, with the equation . Notice how many times the graph crosses the x-axis.
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Interactive Graph: Graph of 5th-Degree Polynomial
Graph of a polynomial function of a degree 5, with the equation . Notice how many times the graph crosses the x-axis.
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Polynomial and Rational Functions
Interactive Graph: Graph of 2nd-Degree Polynomial
Graph of a polynomial function of a degree 2, with the equation . Notice how many times the graph crosses the x-axis.
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Interactive Graph: Graph of 6th-Degree Polynomial
Graph of a polynomial function of a degree 6, with the equation . Notice how many times the graph crosses the x-axis.
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Polynomial and Rational Functions
Interactive Graph: Changing a Constant
A graph of a polynomial with the quadratic equation y=x2+11x+18 and y=2(x2+11x+18). Notice how the intercepts do not change, even when we multiply
the function by a constant.
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Interactive Graph: Graph of Indirect Variation
Graph of indirect variation with the equation y=1/x. This hyperbola shows the indirect variation of variables x and y. Notice what happens when you
change the variants.
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Polynomial and Rational Functions
745 divided by 3
The long division is shown here explicitly to serve as a refresher for more complicated long division of polynomials.
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Polynomial and Rational Functions
Interactive Graph: Example of a Rational Function
Graph of the rational function . This function has three x-intercepts.
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Polynomial and Rational Functions
Illustration of Gay-Lussac's Law, derived from the Combined Gas Law
A constant amount of gas will exert pressure that varies directly with temperature. In this illustration, volume is held constant by an increased mass
weighing down the lid of the container. If not for that extra mass, the lid would raise, increasing the volume and relieving the pressure.
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Polynomial and Rational Functions
Interactive Graph: Plots of Quadratic Equations
Graph of quadratic equations, changing either a, b, or c in the typical equation . Compare all the graphs to the red function, . Varying each coefficient
separately include (blue), (green), and (purple).
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Polynomial and Rational Functions
Curve Fitting
Polynomial curves generated to fit points (black dots) of a sine function: The red line is a first degree polynomial; the green is a second degree; the
orange is a third degree; and the blue is a fourth degree.
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Polynomial and Rational Functions
FOIL Method Diagram
start by multiplying the First terms, then the Outside terms, then the Inside terms, and finally the Last terms. Often, the outside and inside terms can
eventually be added together. It is important to understand this method, in order to be able to perform it in reverse.
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Interactive Graph: Rule of Signs
Graph of . We can graphically see there are two solutions to this polynomial. This still fits with the rule of signs, as -1 is a negative root twice in the
equation.
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Polynomial and Rational Functions
Polynomial long divion
For explanations of each step, see the text.
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Polynomial and Rational Functions
Interactive Graph: Multiplying Fractions
Graph of a polynomial with the quadratic equation of . We can graph this equation, and in doing so see where it intercepts the y axis, as a means of
checking our solutions to this problem.
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Interactive Graph: Finding Rational Solutions
Graph of . One can also use the Rational Zeros Theorem to narrow down the candidates for solutions, then look to see which one is represented by the
graphical form of the equation.
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Interactive Graph: Graph of 3rd-Degree Polynomial
Graph of a polynomial function of a degree 3, with the equation .
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Interactive Graph: Graph of a Rational Polynomial
Graph of a rational polynomial with the equation . For x-values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
For x values that are zeros for the denominator polynomial, the rational function is undefined, with a vertical asymptote forming instead.
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Polynomial and Rational Functions
Example of a quadratic/linear rational function
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Interactive graph: domain of a function
Graph of a rational polynomial with equation . The domain of this function is all not equal to +2 or -2.
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The Maximum Modulus Principle
A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As predicted by the fundamental theorem, the
maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).
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Asymptotes
The graph of a function with a horizontal, vertical, and oblique asymptote.
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The steps of synthetic division
Follow along each step in the text.
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Polynomial and Rational Functions
Describe the end behavior of f(x)=3x3-2x2+x-7.
A) The function inclines to the left and declines to the right.
B) The function inclines both to the left and to the right.
C) The function declines both to the left and to the right.
D) The function declines to the left and inclines to the right.
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Polynomial and Rational Functions
Describe the end behavior of f(x)=3x3-2x2+x-7.
A) The function inclines to the left and declines to the right.
B) The function inclines both to the left and to the right.
C) The function declines both to the left and to the right.
D) The function declines to the left and inclines to the right.
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Describe the end behavior of f(x)=-x2-5x+3.
A) The function declines both to the left and to the right.
B) The function declines to the left and inclines to the right.
C) The function inclines to the left and declines to the right.
D) The function inclines both to the left and to the right.
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Polynomial and Rational Functions
Describe the end behavior of f(x)=-x2-5x+3.
A) The function declines both to the left and to the right.
B) The function declines to the left and inclines to the right.
C) The function inclines to the left and declines to the right.
D) The function inclines both to the left and to the right.
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The graph of f(x)=(x-2)(x+9)(x+5) has zeros at:
A) 2, -9, and -5
B) -2, 9, and 5
C) 0 and 90
D) 0 and -90
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Polynomial and Rational Functions
The graph of f(x)=(x-2)(x+9)(x+5) has zeros at:
A) 2, -9, and -5
B) -2, 9, and 5
C) 0 and 90
D) 0 and -90
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Polynomial and Rational Functions
Find the zeros of the following function: f(x)=(x+3)2(x-2).
A) 3 and -2
B) -3, 3, and -2
C) -3 and 2
D) -3, 3, and 2
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Polynomial and Rational Functions
Find the zeros of the following function: f(x)=(x+3)2(x-2).
A) 3 and -2
B) -3, 3, and -2
C) -3 and 2
D) -3, 3, and 2
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What is one advantage of using rational functions as models
rather than polynomial functions?
A) Rational functions can effectively model asymptotes.
B) All of these answers.
C) Rational functions can take on a greater range of shapes.
D) Rational functions are more accurate with predictions both inside and
outside the range of data.
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Polynomial and Rational Functions
What is one advantage of using rational functions as models
rather than polynomial functions?
A) Rational functions can effectively model asymptotes.
B) All of these answers.
C) Rational functions can take on a greater range of shapes.
D) Rational functions are more accurate with predictions both inside and
outside the range of data.
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Polynomial and Rational Functions
Without actually graphing it, what do we know to be true about the
following polynomial: f(x)=x2+3x+5?
A) All of these answers.
B) It crosses the x-axis two times.
C) It is continuous.
D) It passes the vertical line test.
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Polynomial and Rational Functions
Without actually graphing it, what do we know to be true about the
following polynomial: f(x)=x2+3x+5?
A) All of these answers.
B) It crosses the x-axis two times.
C) It is continuous.
D) It passes the vertical line test.
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Polynomial and Rational Functions
A polynomial graph crosses the x-axis four times. Which of the
following polynomials could represent this graph?
A) f(x)=3x4-6x5+2x-9
B) f(x)=x3+4x2-3x4
C) f(x)=4x2+4x+4
D) f(x)=2x2+3x3+x
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Polynomial and Rational Functions
A polynomial graph crosses the x-axis four times. Which of the
following polynomials could represent this graph?
A) f(x)=3x4-6x5+2x-9
B) f(x)=x3+4x2-3x4
C) f(x)=4x2+4x+4
D) f(x)=2x2+3x3+x
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A continuous function:
A) has no holes.
B) all of these answers.
C) has no asymptotes.
D) is smooth.
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Polynomial and Rational Functions
A continuous function:
A) has no holes.
B) all of these answers.
C) has no asymptotes.
D) is smooth.
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If f(2)=2 and f(10)=25, and f(x) is continuous from 2 to 10, then the
intermediate value theorem tells us:
A) that somewhere between 2 and 10, f(x) must equal 20.
B) that f(5)=16.
C) that f(x) crosses the x-axis between 2 and 10.
D) that the function is continuous over all x-values.
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Polynomial and Rational Functions
If f(2)=2 and f(10)=25, and f(x) is continuous from 2 to 10, then the
intermediate value theorem tells us:
A) that somewhere between 2 and 10, f(x) must equal 20.
B) that f(5)=16.
C) that f(x) crosses the x-axis between 2 and 10.
D) that the function is continuous over all x-values.
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What is x3+7x2-36 divided by x2+4x-12?
A) x+3
B) x-3
C) x+6
D) x-6
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Polynomial and Rational Functions
What is x3+7x2-36 divided by x2+4x-12?
A) x+3
B) x-3
C) x+6
D) x-6
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What is x3+2x2-13x+10 divided by x2+4x-5?
A) x-2
B) x+2
C) x+5
D) x-5
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Polynomial and Rational Functions
What is x3+2x2-13x+10 divided by x2+4x-5?
A) x-2
B) x+2
C) x+5
D) x-5
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Divide x3+2x2-4x+8 by x+2.
A) x2-4
B) x2+4-16/(x+2)
C) x2-4+16/(x+2)
D) x2+4
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Polynomial and Rational Functions
Divide x3+2x2-4x+8 by x+2.
A) x2-4
B) x2+4-16/(x+2)
C) x2-4+16/(x+2)
D) x2+4
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Polynomial and Rational Functions
What is the remainder when 3x3 – 2x2 + 3x – 4 is divided by x – 3
using synthetic division?
A) 68/(x-3)
B) 24/(x-3)
C) 7(x-3)
D) 3/(x-3)
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Polynomial and Rational Functions
What is the remainder when 3x3 – 2x2 + 3x – 4 is divided by x – 3
using synthetic division?
A) 68/(x-3)
B) 24/(x-3)
C) 7(x-3)
D) 3/(x-3)
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Polynomial and Rational Functions
Factor the following polynomial by factoring out the GCF: 2x3-6x.
A) x(2x2-6)
B) 2x(x2-3)
C) 2(x3-3x)
D) 6x(x2-3)
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Polynomial and Rational Functions
Factor the following polynomial by factoring out the GCF: 2x3-6x.
A) x(2x2-6)
B) 2x(x2-3)
C) 2(x3-3x)
D) 6x(x2-3)
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Polynomial and Rational Functions
Factor the following polynomial: x2-81.
A) (x-9)(x+9)
B) (x-9)(x-9)
C) (x+9)(x+9)
D) (x+3)(x+3)(x-3)(x-3)
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Polynomial and Rational Functions
Factor the following polynomial: x2-81.
A) (x-9)(x+9)
B) (x-9)(x-9)
C) (x+9)(x+9)
D) (x+3)(x+3)(x-3)(x-3)
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Polynomial and Rational Functions
Factor the following trinomial: x2+5x-24.
A) (x-8)(x+3)
B) (x-8)(x-3)
C) (x+8)(x-3)
D) (x+8)(x+3)
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Polynomial and Rational Functions
Factor the following trinomial: x2+5x-24.
A) (x-8)(x+3)
B) (x-8)(x-3)
C) (x+8)(x-3)
D) (x+8)(x+3)
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Polynomial and Rational Functions
The fundamental theorem of algebra states that:
A) all non-constant single-variable polynomial with complex coefficients
has at least one complex root.
B) all non-constant single-variable polynomial with complex coefficients
has at least 2 complex roots.
C) all non-constant double-variable polynomial with complex coefficients
has at least one complex root.
D) every constant single-variable polynomial with complex coefficients
has at least two complex roots.
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Polynomial and Rational Functions
The fundamental theorem of algebra states that:
A) all non-constant single-variable polynomial with complex coefficients
has at least one complex root.
B) all non-constant single-variable polynomial with complex coefficients
has at least 2 complex roots.
C) all non-constant double-variable polynomial with complex coefficients
has at least one complex root.
D) every constant single-variable polynomial with complex coefficients
has at least two complex roots.
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Polynomial and Rational Functions
Which of the following represents a polynomial with zeros x=3 and
x=-4?
A) 2x2+2x-24
B) 2x2+x-12
C) x2+x+12
D) x2+2x-12
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Polynomial and Rational Functions
Which of the following represents a polynomial with zeros x=3 and
x=-4?
A) 2x2+2x-24
B) 2x2+x-12
C) x2+x+12
D) x2+2x-12
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Polynomial and Rational Functions
Which of the following polynomials does NOT have zeros x=1 and
x=6?
A) x2-7x+6
B) 2x2-14x+12
C) 3x2-21x+24
D) 5x2-35x+30
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Polynomial and Rational Functions
Which of the following polynomials does NOT have zeros x=1 and
x=6?
A) x2-7x+6
B) 2x2-14x+12
C) 3x2-21x+24
D) 5x2-35x+30
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Polynomial and Rational Functions
Find the zeros of the following function: y=2x2+4x-30.
A) x=3 and x=-5
B) x=-3 and x=5
C) x=6 and x=-10
D) x=-6 and x=10
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Polynomial and Rational Functions
Find the zeros of the following function: y=2x2+4x-30.
A) x=3 and x=-5
B) x=-3 and x=5
C) x=6 and x=-10
D) x=-6 and x=10
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Polynomial and Rational Functions
A polynomial with degree five can have at most how many zeros?
A) 5
B) 4
C) 0
D) 6
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Polynomial and Rational Functions
A polynomial with degree five can have at most how many zeros?
A) 5
B) 4
C) 0
D) 6
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Polynomial and Rational Functions
Which of the following lists all possible rational roots of x4+2x37x2-8x+12?
A) ±1, ±2, ±3, ±4, ±6, ±12
B) ±1, ±12
C) ±2, ±3, ±4, ±6
D) ±1/2, ±6/5, ±12/11
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Polynomial and Rational Functions
Which of the following lists all possible rational roots of x4+2x37x2-8x+12?
A) ±1, ±2, ±3, ±4, ±6, ±12
B) ±1, ±12
C) ±2, ±3, ±4, ±6
D) ±1/2, ±6/5, ±12/11
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Polynomial and Rational Functions
Which of the following lists all possible rational roots of 2x3+3x-5?
A) ±1, ±1/2, ±5, ±5/2
B) ±1, ±5
C) ±2, ±2/5
D) ±1, ±2, ±5
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Polynomial and Rational Functions
Which of the following lists all possible rational roots of 2x3+3x-5?
A) ±1, ±1/2, ±5, ±5/2
B) ±1, ±5
C) ±2, ±2/5
D) ±1, ±2, ±5
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Polynomial and Rational Functions
What is the maximum number of positive roots of x5+_______x423x_______-27x2+166x-120?
A) 4
B) 3
C) 5
D) 2
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Polynomial and Rational Functions
What is the maximum number of positive roots of x5+_______x423x_______-27x2+166x-120?
A) 4
B) 3
C) 5
D) 2
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Polynomial and Rational Functions
What is the maximum number of negative roots of x5+3x4_______3x3-_______7x_______+166x-120?
A) 3
B) 2
C) 4
D) 1
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Polynomial and Rational Functions
What is the maximum number of negative roots of x5+3x4_______3x3-_______7x_______+166x-120?
A) 3
B) 2
C) 4
D) 1
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Polynomial and Rational Functions
What is the minimum number of complex roots of the following
polynomial: x3+_______x_______-4x+3?
A) 3
B) 2
C) 1
D) 0
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Polynomial and Rational Functions
What is the minimum number of complex roots of the following
polynomial: x3+_______x_______-4x+3?
A) 3
B) 2
C) 1
D) 0
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Polynomial and Rational Functions
Solve the following inequality: x3-3x2-16x+48>0.
A) (-∞,-4) (3,4)
B) (-4,∞)
C) (-4,3) (4,∞)
D) (3,4)
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Polynomial and Rational Functions
Solve the following inequality: x3-3x2-16x+48>0.
A) (-∞,-4) (3,4)
B) (-4,∞)
C) (-4,3) (4,∞)
D) (3,4)
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Polynomial and Rational Functions
If y varies directly as x, and y = 24 when x = 16, find y when x =
12.
A) 20
B) 16
C) 28
D) 18
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Polynomial and Rational Functions
If y varies directly as x, and y = 24 when x = 16, find y when x =
12.
A) 20
B) 16
C) 28
D) 18
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Polynomial and Rational Functions
If y varies directly as x , and y = 15 when x = 10 , then what is the
value of k, the constant of variation?
A) 9
B) 2/3
C) 3/2
D) 3
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Polynomial and Rational Functions
If y varies directly as x , and y = 15 when x = 10 , then what is the
value of k, the constant of variation?
A) 9
B) 2/3
C) 3/2
D) 3
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Polynomial and Rational Functions
The number of hours, h, it takes for a block of ice to melt varies
inversely as the temperature, t. If it takes 2 hours for a square
inch of ice to melt at 65º, find the constant of proportionality.
A) 32.5
B) 90
C) 130
D) 56.5
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Polynomial and Rational Functions
The number of hours, h, it takes for a block of ice to melt varies
inversely as the temperature, t. If it takes 2 hours for a square
inch of ice to melt at 65º, find the constant of proportionality.
A) 32.5
B) 90
C) 130
D) 56.5
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Polynomial and Rational Functions
In a formula, Z varies inversely as p. If Z is 200 when p = 4, find Z
when p = 10.
A) 800
B) 500
C) 50
D) 80
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Polynomial and Rational Functions
In a formula, Z varies inversely as p. If Z is 200 when p = 4, find Z
when p = 10.
A) 800
B) 500
C) 50
D) 80
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Polynomial and Rational Functions
If y varies directly as x and inversely as z, and y = 24 when x = 48
and z = 4, find x when y = 44 and z = 6.
A) 2
B) 264
C) 132
D) 4
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Polynomial and Rational Functions
If y varies directly as x and inversely as z, and y = 24 when x = 48
and z = 4, find x when y = 44 and z = 6.
A) 2
B) 264
C) 132
D) 4
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Polynomial and Rational Functions
The resistance of a wire varies directly as its length and inversely
as the square of its diameter. A wire with length 200 in and a
diameter of 1/4 in has a resistance of 20 ohms. Find the
resistance in a 500 in wire with the same diameter.
A) 0.000625 ohms
B) 50 ohms
C) 25 ohms
D) 1.25 ohms
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Polynomial and Rational Functions
The resistance of a wire varies directly as its length and inversely
as the square of its diameter. A wire with length 200 in and a
diameter of 1/4 in has a resistance of 20 ohms. Find the
resistance in a 500 in wire with the same diameter.
A) 0.000625 ohms
B) 50 ohms
C) 25 ohms
D) 1.25 ohms
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Polynomial and Rational Functions
Attribution
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/leading-coefficient
• Wikipedia. "Leading term." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Leading%20term
• Wikipedia. "Coefficient." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Coefficient
• Wikipedia. "Intermediate value theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Intermediate_value_theorem
• Wikipedia. "Continuous function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Continuous_function
• Wiktionary. "interval." CC BY-SA 3.0 http://en.wiktionary.org/wiki/interval
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//psychology/definition/continuous
• Wikipedia. "Polynomial." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial
• Wikipedia. "Real number." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Real_number
• Wikipedia. "Quadratic equation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Quadratic_equation
• Wikipedia. "Complex conjugate root theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Complex_conjugate_root_theorem
• Wiktionary. "root." CC BY-SA 3.0 http://en.wiktionary.org/wiki/root
• Wiktionary. "real number." CC BY-SA 3.0 http://en.wiktionary.org/wiki/real+number
• Wikipedia. "Polynomial." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial
• Wiktionary. "term." CC BY-SA 3.0 http://en.wiktionary.org/wiki/term
• Wiktionary. "polynomial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/polynomial
• Wiktionary. "indeterminate." CC BY-SA 3.0 http://en.wiktionary.org/wiki/indeterminate
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Polynomial and Rational Functions
• Wikipedia. "Rational number." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Rational_number
• Wiktionary. "irrational number." CC BY-SA 3.0 http://en.wiktionary.org/wiki/irrational+number
• Wiktionary. "quotient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/quotient
• Wikipedia. "Polynomial remainder theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial_remainder_theorem
• Wikipedia. "Synthetic division." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Synthetic_division
• Wiktionary. "polynomial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/polynomial
• Wiktionary. "remainder." CC BY-SA 3.0 http://en.wiktionary.org/wiki/remainder
• Connexions. "Finding the Domain of Simple Rational Functions." CC BY 3.0 http://cnx.org/content/m13352/latest/
• Wikipedia. "Rational function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Rational_function
• Wiktionary. "denominator." CC BY-SA 3.0 http://en.wiktionary.org/wiki/denominator
• Wiktionary. "rational function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/rational+function
• Wiktionary. "domain." CC BY-SA 3.0 http://en.wiktionary.org/wiki/domain
• Wikipedia. "Descartes' rule of signs." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Descartes'_rule_of_signs
• Wikipedia. "Descartes' rule of signs." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Descartes'_rule_of_signs
• Wiktionary. "root." CC BY-SA 3.0 http://en.wiktionary.org/wiki/root
• Wiktionary. "sign." CC BY-SA 3.0 http://en.wiktionary.org/wiki/sign
• Wikipedia. "Inequality (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inequality_(mathematics)
• Wiktionary. "inequality." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inequality
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/zero
Free to share, print, make copies and changes. Get yours at www.boundless.com
Polynomial and Rational Functions
• Wiktionary. "polynomial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/polynomial
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/zero
• Connexions. "Factoring." CC BY 3.0 http://cnx.org/content/m18227/latest/?collection=col10624
• Wikipedia. "Asymptote." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Asymptote
• Wiktionary. "rational function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/rational+function
• Wiktionary. "oblique." CC BY-SA 3.0 http://en.wiktionary.org/wiki/oblique
• Wiktionary. "asymptote." CC BY-SA 3.0 http://en.wiktionary.org/wiki/asymptote
• Wikipedia. "Polynomial." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial
• Wiktionary. "inequality." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inequality
• Wikipedia. "Polynomial long division." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial_long_division
• Wikipedia. "Polynomial long division." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Polynomial_long_division
• Connexions. "Dividing Polynomials." CC BY 3.0 http://cnx.org/content/m18299/latest/?collection=col10624
• Wiktionary. "quotient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/quotient
• Wiktionary. "dividend." CC BY-SA 3.0 http://en.wiktionary.org/wiki/dividend
• Wiktionary. "divisor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/divisor
• Connexions. "Rational function." CC BY 3.0 http://cnx.org/content/m15293/1.10/
• Wiktionary. "denominator." CC BY-SA 3.0 http://en.wiktionary.org/wiki/denominator
• Wiktionary. "numerator." CC BY-SA 3.0 http://en.wiktionary.org/wiki/numerator
• Wiktionary. "rational function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/rational+function
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Polynomial and Rational Functions
• Wikipedia. "Fundamental theorem of algebra." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
• Wikipedia. "Maximum modulus principle." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Maximum_modulus_principle
• Wiktionary. "multiplicity." CC BY-SA 3.0 http://en.wiktionary.org/wiki/multiplicity
• Wikipedia. "Proportionality (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Proportionality_(mathematics)
• Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constant
• Wiktionary. "hyperbola." CC BY-SA 3.0 http://en.wiktionary.org/wiki/hyperbola
• Wiktionary. "asymptote." CC BY-SA 3.0 http://en.wiktionary.org/wiki/asymptote
• Wikipedia. "Polynomial and rational function modeling." CC BY-SA 3.0
http://en.wikipedia.org/wiki/Polynomial_and_rational_function_modeling
• Connexions. "Factoring." CC BY 3.0 http://cnx.org/content/m18227/latest/?collection=col10624
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/common-factor
• Wiktionary. "factor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/factor
• Connexions. "Data Concepts -- Variations." CC BY 3.0 http://cnx.org/content/m18281/latest/
• Wikipedia. "Combined gas law." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Combined_gas_law
• Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constant
• Wikipedia. "directly proportional." CC BY-SA 3.0 http://en.wikipedia.org/wiki/directly%20proportional
• Wikipedia. "Rational root theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Rational_root_theorem
• Wiktionary. "coprime." CC BY-SA 3.0 http://en.wiktionary.org/wiki/coprime
• Wikipedia. "Euclid's lemma." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Euclid's%20lemma
• Wikipedia. "Direct variation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Direct_variation#Direct_proportion
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Polynomial and Rational Functions
• Wiktionary. "proportional." CC BY-SA 3.0 http://en.wiktionary.org/wiki/proportional
• Wiktionary. "y-intercept." CC BY-SA 3.0 http://en.wiktionary.org/wiki/y-intercept
• Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constant
• Wikipedia. "Properties of polynomial roots." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Properties_of_polynomial_roots
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/zero
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