4.1 Polynomial Functions
Polynomial – any algebraic function that can be written in the form of axn + axn-1+….ax³
+ ax² + ax + axº
n = nonnegative integer
x – variable
a = coefficient
Characteristics of a Polynomial:
1) All exponents are whole #’s
2) No variable is contained in the denominator
3) No variable is under the radical sign
Types of Polynomials:
a) Constant Polynomial – a polynomial that consists of only a constant term. (ex. 23)
b) Zero Polynomial – the constant polynomial 0
Degree of Polynomials – the exponent of the highest power of x that appears with a
nonzero coefficient. – the nonzero coefficient = leading coefficient
Ex. find the degree of
15x² - x³ + 17 – 11x8 + 63 x5
Linear Functions = first degree polynomials = x + 11
Quadratic functions = second degree polynomials = x² + 1
Cubic Functions = third degree polynomials = x³+ 5
Quartic Functions = fourth degree polynomials = x4 – 3
p. 977 review of operations on polynomials
Polynomial Division - the division algorithm
a) Long division – just like long division of numbers
** must account for missing degrees by adding 0xn to whatever power
Ex. divide 2x4 – 25x² - 30 x + 5 by x – 4
b) synthetic division – use whenever divisor is a first degree polynomial
Ex. x4 + 5x³ + 2x² + 6x – 17 by x + 5
c) Checking polynomial division
Dividend = divisor · quotient + remainder
Division Algorithm – if a polynomial f(x) is divided by a nonzero polynomial h(x) then
there is a quotient polynomial q(x) and a remainder polynomial
r(x) such that:
f(x) = h(x) · q(x) + r(x)
*where either r(x) = 0 or r(x) has degree less than the degree of the
divisor h(x)
Remainders and Factors – if the remainder is 0 when one polynomial is divided by
another polynomial, the divisor and the quotient are factors of the
dividend.
Ex. determine if 3x² - 2 is a factor of 3x³ + 15x² - 2x – 10.
Remainder Theorem – if a polynomial f(x) is divided by x-c, then the remainder is f(c).
** occurs when a polynomial is divided by a first degree polynomial (ex. x-3)
**divisor must be in form of (x-c)
Ex. Find the remainder when x87 – 2x36 + 8 is divided by x + 1
Step 1: divisor is x + 1 so c = -1
Step 2: plug into original equation
(-1)87 – 2(-1)36 + 8
=5
so the remainder is 5
Zeros and Factors
Zeros – if f(x) is a polynomial, then the solutions of the equation f(x) = 0
a) real zero – a zero that is a real number
Leads to:
Factor Theorem – a polynomial function f(x) has a linear factor x – a if and only
if: f(a) = 0
1) if a is a solution of f(x) = 0, then x-a is a factor of f(x)
2) if x-a is a factor of f(x), then a is a solution of f(x) =0
Ex. Show that x-2 is a factor of x³ + 5x² -2x -24 using the factor theorem.
a) find q(x) such that (x-2)q(x) = x³+5x²-2x-24
f(2) = 2³ + 5(2)² - 24
= 0 so that means x-2 is a factor
**now check using the division algorithm – divide (x-2) into (x³+5x²-2x-24)
Will get q(x)=(x²+7x+12) so:
x³+5x²-2x-24 = (x-2)(x²+7x+12) --- f(x) = h(x)·q(x)
Zeros, X-intercepts, Solutions, and Factors of Polynomials:
Let f(x) be a polynomial. If r is a real number that satisfies any of the
following, then r satisfies all of the statements.
1. r is a zero of the function f
2. r is an x-intercept of the graph of the function f
3. x = r is a solution, or root, of the equation f(x) = 0
4. x-r is a factor of the polynomial f(x)
Ex. for f(x) = 2x³ - 7x² -20x + 25 find:
A) the x- intercepts
B) the zeros of f
C) the solutions of f
D) the linear factors of f
Ex. find a polynomial that have -2, 1, and 3 as zeros
Number of Zeros : a polynomial of degree n has at most n distinct real zeros
4.2 Real Zeros
** Finding the real zeros of a polynomial f(x) is the same as solving the polynomial
equation f(x) = 0
Rational Zero Test – applied it a polynomial has integer coefficients
A) If a rational number p/q (written in lowest terms is a zero of the
polynomial function) .. f(x) = anxn + …+ a1x + a0
B) p is a factor of the constant term a0
C) q is a factor of the leading coefficient an
**Numerator – is a factor of the constant term
**Denominator – is a factor of the leading coefficient
d) Must find all the #’s that satisfy the above conditions and a list
of rational zeros is produced – must be evaluated at each #
Ex. Checking with a calculator: Find the rational zeros of f(x) = 2x4-7x³-19x²-3x + 7
Possible rational zeros are: p/q = ±1, ±7, ±½, ±7/2
a) Must check to see which of these are the real zeros, so put into calculator and
Graph- the ones that come up on the screen are the only ones really to test
b)then check the table function to see which ones actually have a zero.
Zero and Factor Theorem
a) once some zeros of the polynomial have been found – the Factor Theorem can be
used to factor the polynomial
b) divide the known factors into the poly. using synthetic division,
then find the remaining factors using synthetic division (down to x²term)
c) whole goal is to reduce the polynomials down to all its factors
Descartes Rule of signs:
1. Check for positive real zeros by counting the number of sign changes of the given
function. It will then decrease by intervals of 2.
2. For negative real zeros check the number of sign changes of (-x) when inserted
into the function.
Ex. P(x) = -2x4 + x³ -7x² + 4x +1
Number of positive zeros:
Number of negative zeros:
Irreducible – when a polynomial can’t be written as the product of polynomials of lesser
Degrees
A) Completely factored over the set of real numbers – when the product is
written as the product of irreducible factors with real coefficients.
Ex. Factor f(x) x5 -5x4+8x³ -8x²+7x – 3 completely
1) Using the rational zero test, the possible rational zeros are ±1, ±3 – check on calculator
To see what are 0’s = 1, and 3
2) Use synthetic division to make sure x-1 and x-3 are real solutions, and what the
Polynomial that is left after divided into.
3)Try the roots again to see what works on the polynomial that is not yet completely
Factored
Bounds Test:
** Let f(x) be a polynomial with positive leading coefficient.
a) If d>0 and every # in the last row in synthetic division of f(x) by x – d is
Nonnegative, then d is an upper bound for the real zeros of f
b) If c<0 and the #’s in the last row of the synthetic division of f(x) by x-c are
alternately positive and negative, with 0 considered as either, then c is a
lower bound for the real zeros of f.
Ex. Find all real zeros of f(x) = x6 + x³ -8x² -4x + 1
Steps: a) do rational root test - get only ±1
b) graph – see if any real zeros on table,
c) from the graph then, look at what seems to be an upper and lower bound
and test using synthetic division
d) do the bounds test
e) if both pass then shows that all the zeros fall in between the bounds
f) use calculator zero finder to find the zeros
Finding Real Zeros of Polynomials
1. Use the Rational Zero test to find all the rational zeros of f
2. Write f(x) as the product of linear factors, one for each rational zero, and another
factor g(x)
3. If g(x) has a degree 2, find the zeros by factoring or by using the quadratic
formula.
4. If g(x) has degree 3 or greater, use the Bounds Test, if possible, to find the lower
and upper bounds for the zeros of g. Approximate the remaining zeros
graphically.
Ex. Find all the real zeros of f(x) = x7 – 2x6-5x5 + 13x4-10x³+11x²-4x-4
4.3 Graphs of polynomial functions
Basic Polynomial Shape- of the form f(x) = axn (a=constant, n=a nonnegative integer)
a) when degree of a polynomial function in the form f(x) = axn is odd – graph
will have the general hyperbolic form - or look reflected across x or y axis
b) when the form f(x) = axn is even, the graph is parabolic, and when negative is
the reflection across the x-axis
Ex.
Properties of General Polynomial Functions:
1. Continuity – a graph that is an unbroken curve, with no jumps, gaps, or holes,
and has no sharp corners (on calc – some appear sharp, need to zoom in to see)
2. End Behavior – the shape of a polynomial graph at the far left and far right of the
coordinate plane – when │x│ is very large.
a) when │x│ is large, the graph of a polynomial function closely
resembles the graph of its highest degree term.
b) when a polynomial function has odd degree, one end of its graph
shoots upward, and the other end downward.
c) when a polynomial function has even degree, both ends of its graph
shoot upward or both ends shoot downward
**look at graphs on p. 263/264
3. Intercepts – the graph of a polynomial function of degree n
a) has 1 y-intercept which is equal to the constant term
b) has at most n x-intercepts (can’t have more than the highest degree)
4. Multiplicity – occurs if x – r is a factor that occurs m times in the complete
factorization of a polynomial expression
ex. f(x) = (x+3)²(x+1)(x-1)³ multiplicity of -3 is 2, -1 is 1, and 1 is 3
let c be a zero of multiplicity k or a polynomial f
a) if k is odd, the graph of f crosses the x-axis at c
b) if k is even, the graph of f touches, but does not cross the x-axis at c
ex. find all zeros of f(x) = (x+1)³(x-1)(x-3)². State the multiplicity of each,
whether it touches or crosses the x-axis
5. Local Extrema – refers to either a local maximum or a local minimum – where
graph has peaks and valleys – occurs when graph change from
increasing to decreasing or vice versa
a) a polynomial function of degree n has at most n-1 local extrema
6. Points of Inflection – occurs where the concavity of a graph of a function
changes
a) the graph of a polynomial function of degree n, with n≥2, has at most
n-2 points of inflection
b) The graph of a polynomial function of odd degree, with n>2 has at least 1
Point of inflection
Complete graphs of polynomials:
Ex. find a complete graph of: f(x) = x4 +11x³ + 25x² - 50x -70
4.4 Rational Functions
Rational function – is a function whose rule is the quotient of 2 polynomials such as:
F(x) = 1
x
f(x) = 4x-3
2x+1
f(x) = 2x³ + 5x + 2
x² - 7x +6
Domain of Rational Functions – the set of all real numbers that are not zeros of its
denominators.
Properties of Rational Graphs: - calc’s do a poor job of graphing rational functions
1. Intercepts
a) If f has a y-intercept, it occurs at f(0)
b) X-intercepts – occur at numbers that
1) are zeros of the numerator
2) are not zeros of the denominator
Ex. find the intercepts of:
f(x) = x² - x – 6
x–2
2. Continuity – there are breaks in the graph of the rational function wherever the
function is not defined (that is the zeros)
a) except for these breaks, the graph is a continuous unbroken curve
b) no sharp corners
3. Vertical Asymptotes – a rational function has breaks in its graph at all points
where the function is not defined.
a) occur at every number that is a zero of the denominator but not of the
numerator
b) understanding behavior near these asymptotes is :
The Big-Little Concept: if c is a# far from 0, then 1/c is a # close to 0
: if c is close to 0, then 1/c is far from 0
** check calculator table to confirm this
c) curves approach vertical asymptotes very closely – but never touch it!!, or
crosses it because it is undefined at that point!!
4. Holes – let f(x) = g(x)
h(x) be a rational function and let d denote a 0 of both g and h
 if the multiplicity of d as a zero of g is greater than or equal to its
multiplicity as a zero of h, then the graph of f has a hole at x = d
 Otherwise the graph has a vertical asymptote at x = d
Ex. compare: f(x) = x-3 , f(x) = (x-3)(x-3) , f(x) = x – 3
x-3
x–3
(x-3)(x-3)
Hole vs. Vertical Asymptote
a) Hole –also called removable discontinuity occurs when factors cancel out in
numerator and denominator.
Ex. x² - 4
= (x+2)(x-2) = (x+2)
x–2
(x – 2)
** x²-4 and (x+2) are not the same graph or same values b/c if I plug a 2 in 1st
x–2
one I get 0 and 2nd one I get 4. So I put a hole at that point
Calculator Warnings: -- may show a vertical segment – not an asymptote – will need to
Change your window—also often won’t show holes unless adjust
5
End Behavior – the behavior of a rational function when x is very large
** knowing end behavior makes it easier to determine end behavior of
rational functions in which the degree of the numerator is ≤ degree of
denominator
a) horizontal asymptote –
1) when the degree of leading coefficient in numerator is > degree
of denominator -- then no HA – a slant asymptote
2) when degree of leading coefficient in numerator is = to degree
in denominator – HA = coefficient in front of x
3) when degree of leading coefficient in numerator is < degree of
denominator—then HA = 0
Ex. List the horizontal and vertical asymptotes and describe the end
behavior of:
a) f(x) = -4x + 8
2 – 3x
b) g(x) =
x
x² - 9
b) Slant/Oblique Asymptotes – occur when leading coefficient degree in
numerator > degree in denominator
1) To find, must use polynomial division – and the quotient will
Be the slant/oblique asymptote
Ex. describe the end behavior of : f(x) = x² - x- 6
x+4
c) parabolic asymptote – when divide and quotient is of x²
Ex. describe end behavior of:
F(X) = x³+ x²+3x – 1
x+1
Graphs of Rational Functions:
1. Analyze the function algebraically to find vertical asymptotes, holes, intercepts
2. Determine end behavior of the graph
3. Use above info. to select appropriate viewing window to have complete graph.
4.5 Complex Numbers
1. Complex # system contains all real numbers!!
2. +/-/x/÷ of complex #’s obey the same rules of arithmetic, except one:
**the exponent laws hold for integer exponents, but not necessarily for fractional
ones. **
3. Complex # system contains a number, denoted i such that i² = -1
4. Every complex # can be written in standard form:
a + bi
where a and b are real #’s
5. a + bi =c + di if and only if a=c and b=d
Sums of real and imaginary numbers in form a + bi = complex #s
Ex. 5 + 2i , 7 – 4i
** every real # a is a complex # because it can be written a + 0i
**every imaginary # b is complex because it can be written 0 + bi
Equating 2 Complex #’s
Ex. find x and y if -4x + 5i = 8 – 6yi
Operations on complex numbers:
1. Addition/Subtraction:
Ex. (2 – 3i) + (1 + i) =
Ex. (2 + 3i) – (4 – 2i) =
2. Multiplication:
Ex. 10i(5 + 1/5i) =
Ex. (3 + i)(2 – 5i)=
Powers of i form a cycle – must be one of 4 possible values: i, -1, -i, 1
a) i¹ = i
b) i² = -1
c) i³ = i²·i = -1· i = -i
d) i4 = i² · i² = -1 · -1 = 1
To find higher powers of i – just divide by 4 and match the remainder to one of above
Ex. i73 = (73/4 = 18 with remainder 1) so = i¹ = i
Ex. i86=
Conjugates: of a + bi is a-bi and the conjugate of a-bi is a + bi
Ex. conjugate of 3+4i =
Ex. conjugate of -3i
**Product of conjugate pairs is a real #**
Ex. (3+4i)(3-4i)
Quotients of Complex Numbers -- need to use conjugates
Ex. -2+5i
1-2i
Square roots of negative #’s -- let b be a positive real # then √-b = i√b
Ex.
a) √-5 =
b) 2 - √11
5
c) (1-√-16)(5 + √-36)
Every quadratic equation with real coefficients has solutions in the complex # system
Ex. x² =-25
x = ±5i
Ex. Find all solutions to 3x²-5x +13 = 0
(use quadratic formula)
Ex. x³ = 64
4.6 The Fundamental Theorem of Algebra
** all the things we did in section 4.1 (factor theorem. Division algorithm) all apply to
complex numbers.
Ex. Is (x-i) a factor of f(x) = x² + (-1+i)x + (2+i)
Fundamental Theorem of Algebra: Every nonconstant polynomial has a zero in the
complex number system.
Factorization over Complex Numbers: states that every polynomial of degree n>0 can
be written as the product of n linear factors: f(x) = a (x-c1)(x-c2)(x-c3)…. Where a is the
leading coefficient.
Ex. Find a polynomial f(x) of degree 4 such that -3, -1 and 2 are zeros, 2 is a multiplicity
of 2 and f(1) = 32
( steps: need to solve for a first using above formula)
Remember that the conjugate of a complex # a+bi is a-bi.
**We usually write a complex # by the abbreviation z
** The conjugate is written as z and read “z bar”
Ex. find all the zeros of x² - 4x +5
Conjugate Zero Theorem: Let f(x) be a polynomial with real coefficients. If the
complex number z is a zero of f, then its conjugate z is also a zero of f. (Because the i’s
end up cancelling out, and you get real numbers.)
Ex. Find a polynomial with real coefficients whose zeros include the numbers -3 and 2-i
Factorization over the Real Numbers: Every nonconstant polynomial with real
coefficients can be factored as a product of linear and irreducible quadratic polynomials
with real coefficients in such a way that the quadratic factors, if any, have no real zeros.
Ex. Completely factor x4 – 4x3 – 2x² + 28x – 35 over the set of #’s given that 2-i is a zero
of f.