Final Exam Review
Chapter 1: Number Systems and Fundamental Concepts of Algebra
Scientific Notation: Numbers written as a x 10n where 1 < |a| < 10 and n is an integer
If n is negative, the number is small; if n is positive, the number is large
n tells you the number of places to move the decimal point
If a is negative, the original number is negative
Properties of Exponents: A negative sign is not part of the base unless grouping is used:
(-3)2 = 9; the base is -3
-32 = -9; the base is 3
Review the properties
Be careful to not apply an exponent property to a constant in a base; these are properties of
exponents: (4x3y2z5)3
Any base to the 0 power equals 1
Be careful with polynomial bases, (2x + y)2 must be expanded (2x + y)(2x + y) and then multiplied out
because your base contains terms rather than factors.
We do not leave a negative exponent in the final result.
Properties of Radicals: Look at the index and the radicand to determine if the expression is defined. If
the index is even, the radicand must be greater than or equal to 0 to be defined in the real number
system. When taking roots, you may need to use absolute value to ensure your root is the principal
(positive) root.
If the index is odd, the radical is defined for all real numbers; a negative radicand yields a negative root
and a positive radicand a positive root.
A radicand is simplified when:
1. The radicand contains no factor with an exponent greater than or equal to the index.
2. The radicand contains no fractions.
3. The denominator does not contain a radical. We rationalize the denominator by multiplying the
numerator and denominator by a radical that will make the denominator a perfect root. If the
denominator is a binomial containing a radical we multiply by its conjugate; i.e. if the
denominator is 8 + √𝑥 we multiply the numerator and denominator by its conjugate: 8 - √𝑥.
You can add “like” radicals – the indices and radicand must be the same; you can multiply (in radical
form) as long as the indices are the same.
𝑚
𝑛
𝑛
𝑚
Rational numbers can also be exponents: 𝑎 𝑛 = √𝑎𝑚 = √𝑎 .
Factoring:
Always factor out a GCF first, then:
Binomials can possibly factored as one of the following:
Difference of Two Squares: A2 – B2 = (A – B)(A + B)
Difference of Two Cubes: A3 – B3 = (A – B)(A2 + AB + B2)
Sum of Two Cubes: A3 + B3 = (A + B)(A2 – AB + B2)
4-term polynomials can possibly be factored by grouping (recall you may have to rearrange the terms)
Trinomials: may be factored using the grouping method or trial and error
A polynomial that cannot be factored is said to be prime.
The Complex Number System: a + bi is standard form of a complex number.
𝑖 = √−1
i 2 = -1
Powers of i repeat in a pattern: i, -1, -i, 1. To simplify a power of i, divide the exponent by 4 and raise i
to the power of the remainder. Then, replace that power of i appropriately.
Never leave i in the denominator of a fraction. You must rationalize the denominator. This may require
the use of the complex conjugate (i.e. the complex conjugate of 8 – i is 8 + i).
Chapter 2: Equations and Inequalities in One Variable
3 types of equations: identities (have an infinite number of solutions R), contradictions (solutions are
the empty set Ø – there are no solutions), and conditional.
A solution is the value(s) of the variable(s) that make the equation a true statement.
Before beginning the solution process, identity what type of equation you have, then think and solve.
Linear Equations in one variable: simplify and “undo” what’s been done to the variable using addition,
subtraction, multiplication and/or division on both sides of the equation until the variable is isolated.
Absolute value equations: think of distance from 0 on the number line:
If c > 0, then |ax+b| = c is equivalent to ax+b = c or ax+b = -c. Then solve each equation for x.
if c = 0, then |ax+b | = 0 and ax+b = 0
if c < 0, then |ax+b | = c has solution set Ø. Absolute value cannot be negative
Always isolate absolute value first – then set up your “cases” (if necessary)
You should also be able to solve a formula for a specified variable.
Linear inequalities: solve exactly like linear equations except when multiplying or dividing by a negative
number you reverse the inequality (the sense of the inequality). These have an infinite number of
solutions and solutions are typically given in interval notation or on a number line graph.
Compound inequalities can be written in the form a < x < b or using the conjunction “and” or “or”;
intersection is related to the word “and” whereas union is related to the word “or”.
Absolute value inequalities: Again, think of the distance from 0 on the number line and set up your
cases.
|ax+b| < c implies -c < ax+b < c
|ax+b | > c implies ax+b < -c or ax+b > c
Quadratic equations: second degree. MAY be solved by factoring or utilizing the square root property;
can also solve any quadratic equation using the quadratic formula or by completing the square. You may
or may not need to simplify first to get it in standard form.
Rational equations (variable in the denominator of a fraction): multiply both sides of the entire
equation by the LCD to remove all fractions, then solve for the variable.
Also in this section is the simplification of rational expressions: adding, subtracting (must have a
common denominator), multiplying, and dividing along with the simplification of a complex fraction (see
Examples 1-4 in Section 2.5).
Radical equations (variable under a radical sign): isolate the radical, then raise both sides to the
appropriate power to remove the radical. Then solve the resulting equation. Recall you must check to
see if there are extraneous solutions (i.e., the value doesn’t make the original equation true).
Chapter 3: Linear Equations and Inequalities in Two Variables
Linear equations in two variables have an infinite number of ordered pair solutions.
To find the y-intercept: let x = 0 and solve for y. It is the point where the graph intersects the y-axis.
To find the x-intercept: let y = 0 and solve for x. It is the point where the graph intersects the x-axis.
Note: you should be able to find the intercepts of any equation – not just linear equations
The standard form of a linear equation is ax + by = c. Any equation that can be put into that form is a
linear equation.
x = c is a vertical line; its slope is undefined
y = c is a horizontal line; its slope is 0
𝑦 −𝑦
The slope of a line: 𝑚 = 𝑥2 − 𝑥1 or
2
1
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
Slope-intercept form is y = mx + b: If a linear equation in two variables is solved for y, it is in the form
y = mx + b, where m is the slope of the line and the y-intercept is (0,b).
Two lines are parallel if their slopes are equal.
Two lines are perpendicular if their slopes are negative (opposite) reciprocals of one another.
The point-slope form of a line is y – y1 = m(x – x1) where m is slope and (x1, y1) is a point on a line. This
form is useful when finding equations of lines.
Chapter 4: Relations, Functions, and Their Graphs
A relation is a set of ordered pairs.
A function is a set of ordered pairs such that each x-value is paired with a unique y-value (that is, all the
x-coordinates are different in the ordered pairs). The graph of a function passes the vertical line test.
Function notation: f(x) is used in place of the variable y.
The domain of a relation is the set of all x-values for which the relation is defined.
The range of a relation is the set of all y-values for which the relation is defined.
Parabolas: The graph of the function g(x) = a(x – h)2 + k is a parabola whose vertex is at (h, k). The
parabola is narrower (stretched) than f(x) = x2 if |a| > 1, and is broader (compressed) than f(x) = x2 if 0 <
|a| < 1. The parabola opens upward if a is positive and downward if a is negative.
With this and other functions, you should be able to identify shifts (both horizontally and vertically),
reflections, and stretches and compressions. (This is what we did in Section 4.4.)
You should also be familiar with piecewise-defined functions (Example 4, page 272).
Variation: y varies directly as x: y = kx where k is the constant of proportionality.
𝑘
y varies inversely as x: y = 𝑥 where k is the constant of proportionality.
You should be able to evaluate functions at a given value of the variable as well as add, subtract,
multiply, divide, and compose two functions (Section 4.5).
You should be able to find the inverse of a function.
If a function passes a horizontal line test, its inverse is also a function. If a function does not pass the
horizontal line test its inverse is simply a relation – not a function.
We represent inverses as f-1 or f-1(x).
Chapter 5: Polynomial Functions
We looked at dividing one polynomial by another and used long division and synthetic division.
Synthetic division can only be used when the divisor is of the form x – k.
If there is no remainder, the divisor is a factor of the polynomial. If there is a remainder AND the divisor
is of the form x – k, then p(k) = remainder (the value of the polynomial p for x = k equals the remainder).
Section 8.1: Solving Systems by Substitution and Elimination
The solution of a consistent system of equations is the ordered pair that satisfies all equations in the
system. We used both the Substitution and the Elimination Methods to solve these.
EXAM
You will have 2 hours to take the exam. Manage your time carefully. Do NOT spend too much time on
any single problem.
Your exam will consist of 50 multiple-choice questions. You will put the letter of your answer on the
blank provided on the test and bubble in a scantron sheet with the correct answer. Please make sure
you have a #2 pencil with eraser with you.
I suggest you work each problem as usual. Once you have your answer, THEN look at the choices.
Hopefully, your answer is one of the choices. If not, reexamine the problem thinking about the
procedure or ways to check the result.
Work the textbook problems I gave to help you study for the Final Exam. Also, look over the “Review for
Final Exam” powerpoint slides on my website.
I have enjoyed having all of you in class. Best wishes on the final!