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
Properties of Exponents: A negative 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 on page 28
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, i.e. (2x + y)2 must be expanded (2x + y)(2x + y) and then simplified
You cannot just “bring in the power” because your base contains terms rather than factors.
Traditionally we do not leave a negative exponent in a result – and remember the order of operations is
in effect.
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 extracting 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 real numbers; moreover, 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 denominators 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 +
we multiply the numerator and denominator by its conjugate: 8 -
.
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.
Rational numbers can also be exponents:
=
. Also,
.
It is sometimes helpful to use exponential form rather than radical form to simplify expressions (see
Examples 7 and 8 on pages 45 and 46).
Factoring: Factoring is the process of writing an indicated sum/difference of terms as a product of
factors.
Always factor out a GCF first, then look:
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 ac-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. We have expanded the
real number system.
=i
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), contradictions (solutions are the
null set – there are no solutions), and conditional.
A solution/root/zero 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 transform the equation by addition, subtraction,
multiplication and division until the variable is isolated.
Absolute value equations: think of distance from 0 on the number line:
If c > 0, then |x| = c is equivalent to x = c or x = -c
if c = 0, then |x| = 0 and x = 0
if c < 0, then |x| = c is Ø Absolute value can not be negative
Always isolate absolute value first – then set up your “cases” (if necessary)
You should also be able to solve an equation for a specified variable (Ex. 5, page 93).
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 simplify! LOOK and THINK! Consider the discriminant: x2 + 6x + 9; x2 + 6x – 8; x2 + 6x + 10. It
lets you predict the type of solutions you will have.
Higher degree – quadratic like equations: may require us to use a substitution (see Example 1 on page
124); some can be solved by factoring. Again, LOOK and THINK!
Rational equations (variable in the denominator of a fraction): multiply through the entire equation by
the LCD to remove all fractions
Radical equations (variable under a radical sign): isolate the radical, then raise both sides to an
appropriate power to remove the radical. This may take several steps.
Linear inequalities: solve exactly like linear equations except when multiplying or dividing by a negative
number you reverse the inequality sign (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 connector “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” (if necessary).
|x| < a implies -a < x < a
|x| > a implies x > a or x < - a
Also in this chapter 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).
Chapter 3: Linear Equations and Inequalities in Two Variables
Linear equations in two variables have an infinite number of ordered pair solutions.
The distance formula: d =
symbol. Distance is always a positive number.
. Remember, the radical sign is a grouping
The midpoint formula:
. These are the coordinates of the midpoint of the segment
joining two given points.
To find x-intercepts: let y = 0 and solve for x. It is the point where the graph intersects the x-axis.
To find y-intercepts: let x = 0 and solve for y. It is the point where the graph intersects the y-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, where a and b are not both equal to 0. 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
The slope of a line is its amount of inclination and can be represented numerically:
or
or
If a linear equation in two variables is solved for y, it is in the form y = mx + b, where m, the coefficient
of x is the slope of the line and b, the constant term, is the y-intercept. This is called slope-intercept
form and makes it very easy to graph the equation.
Two lines are parallel if their slopes are equal; if the slopes of two lines are equal, the lines are parallel.
Two lines are perpendicular if their slopes are negative reciprocals of one another (their product is -1); if
the slopes of two lines are negative reciprocals (the product is -1) then the lines are perpendicular.
The point-slope form of a line is y – y1 = m(x – x1) where m is slope and (x1, y1) are the coordinates of a
point on a line. This form is useful when writing equations of lines.
To graph linear inequalities in two variables, we graph the boundary line (either as a solid line or a
dotted line, depending on the inequality sign). That boundary line separates the plane into two halfplanes. We test a point in one of the half-planes in the original inequality. If it tests true, we shade that
side of the boundary line because all points in that half-plane are solutions of the inequality. If it tests
false, we do not shade that side of the half-plane but rather the other half-plane.
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. The graph of a function satisfies the vertical line test.
Every function is a relation, however, every relation is not a function.
Function notation: f(x)
The domain of a relation is the set of all x-values for which the relation is defined. We look at graphs
from left-right to determine domain.
The range of a relation is the set of all y-values for which the relation is defined. We look at graphs from
low-high.
Parabolas: The graph of the function g(x) = a(x – h)2 + k where a, h, and k are real numbers and a ≠ 0, 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.
You should be familiar with the graphs of functions in the following forms and aware of how constants
affect the graph(s) as well as the manner in which a negative affects the graph:
f(x) = axn for n both odd and even.
f(x) =
for n both odd and even.
f(x) =
for n both odd and even (these are your root functions).
f(x) = a|x|
(pages 266-271 illustrate these functions)
With all of these functions, you should be able to identify shifts (both horizontally and vertically) as well
as 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. When x increases, y
also increases.
y varies inversely as x: y =
where k is the constant of proportionality. When x increases, y decreases.
(matching functions with graphs – #30-37 page 276 – is a good exercise)
Even/Odd Functions:
An even function has a graph that is symmetric to the y-axis and f(x) = f(-x)
An odd function has symmetry about the origin and –f(x) = f(-x)
If a relation has x-axis symmetry it is not a function.
Caution: do not get even/odd functions confused with the degree of a polynomial. Just because the
degree is even does not mean the function is necessarily even.
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. Remember, composition is not commutative.
You should be able to find the inverse of a function (replace x with y and y with x ) in the definition of
the function. It, of course, reverses the domain and the range.
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
The zero of a function, its root(s) and solution(s) are all synonymous terms.
The degree of a polynomial is the degree of its highest term.
To solve a polynomial inequality, find the x- and y- intercepts. The x-intercepts partition the number line
and you may test points in the various regions to see if the inequality tests true or false. Graphing the
intercepts also helps us graph the function itself. The graphing calculator also is helpful in obtaining a
good graph which should display all the intercepts as well as any lows or highs (maximums and
minimums on given intervals).
You should also be aware of the end behavior of various polynomial functions, i.e. as x→±∞ what
happens to y, f(x).
Look at the matching on page 344, #50-61 – hopefully, we can do these in small groups.
We looked at dividing one polynomial by another – and used methods of long division and synthetic
division. BE CAREFUL – 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 system of equations are the ordered pair(s) that satisfy all equations in the system.
With lines, realize the lines can intersect (one solution), be parallel (no solutions), or one line can lie on
top of another (all ordered pairs of one of the equations are solutions for all equations in the system).
EXAM
You will have 2 hours to take the exam.
Your exam consists of 50 multiple-choice questions. KEEP UP WITH YOUR TIME!!!!!! Do NOT spend too
much time on a single question. You will put the letter of your answer on a blank provided on the test
and bubble in a scantron sheet with the correct answer. Please make sure you have a #2 pencil with you.
I suggest you work a problem, come up with what you believe to be the solution/answer, THEN look at
the choices. Hopefully, your solution is one of the ones specified. If not – redo the problem or THINK.
You are smart people!
Read the summaries (in blue) at the end of each chapter, look over class examples and tests from the
semester.