Math 152B Practice Midterm II
Please do all of the following problems. All work and all answers must be done
on your own paper. Credit earned will be based on the steps that you show that
lead to the final solution. Good Luck!
Problem 1 Find the root:
Sol
Special Products and Factoring Strategies
Review of Three Special Products
Recall the three special products:
1.
Difference of Squares
x2  y2 = (x  y) (x + y)
2.
Square of Sum
x2 + 2xy + y2 = (x + y)2
3.
Square of Difference
x2  2xy + y2 = (x  y)2
Spe
Rational Exponents
Square and Cube Roots as Exponents
We define a1/2 as the nonnegative number such that when you square it, you get a.
Example
91/2 = 3
We define a1/3 as the number such that when you cube it you get a.
Example
81/3 = 2
Rational Roots
We
Math 152B Midterm III Key
Please do all of the following problems. All work and all answers must be done
on your own paper. Credit earned will be based on the steps that you show that
lead to the final solution. Good Luck!
Problem 1:
Write down the quadra
Multiplying and Dividing Radical Expressions
Product Rule for Radicals
Often, an expression is given that involves radicals that can be simplified using rules
of exponents. One such rule is the product rule for radicals
Product Rule for Radicals
Example
E
Multiplication and Division of Rational
Expressions
Simplifying Rational Expressions
We define a Rational Expression as a fraction where the numerator and the
denominator are polynomials in one or more variables.
Examples
A.
x2  y2
is a rational expressi
Radical Equations and Complex Numbers
Radical Equations
If we have an equation with a single radical then we follow the procedure:
Step 1 Isolate the radical so that the radical is alone on the left side of the
equation with everything else on the other s
Complex Fractions and Equations with Rational Expressions
Complex Fractions
First we begin with a complex fraction that contains no variables.
Example
1
5
1
5
2
12 6
2
12
6
=
1
2
1
+
4
2
12 +
3
4
6  10
=
12
3
4
=
3+8
Multiply Numerator and
Denominator by
Formulas and Absolute Value Inequalities
Problem Solving With Formulas
A formula is an equation that relates real world quantities.
Examples
P = 2l + 2w
is the formula for the perimeter P of a rectangle given the length l and width w.
d = rt
is the formul
The Quadratic Formula
Quadratic Formula
Lets complete the square for
ax2 + bx + c
a(x2 + b/a x) + c
1.
2.
b
2a
3.
b2
4a2
b2
b
4.
a( x2 +
b2
x +

a
4a
a[ ( x2 +
4a
b2
x +
)
a
4a
2
)2 
a[ ( x +
] +c
b2
b
)2 
a( x +
4a
4a2
2a
7.
] +c
b2
b
6.
) +c
2
b2
b
Linear Inequalities, Variation, and Solving by
Graphing
Definition of a Linear Inequality
Definition
A linear inequality is an inequality that has one of the four forms below
1.
Ax + By < C
2.
Ax + By > C
3.
Ax + By < C
4.
Ax + By > C
We follow the follow
GCF and Factoring Trinomials
Factoring Out the Greatest Common Factor (GCF)
Consider the example
x (x  3) = x2  3x
We can do the process in reverse:
Look at
x2  3x
and notice that both have a common factor of x.
We can pull out the x terms (use the dis
Math 152B Midterm I
Please do all of the following problems. Credit earned will be based on the steps that you
show that lead to the final solution. Good Luck!
Problem 1: Factor the expression completely:
A) x4  9x2
Solution: x2(x2  9)
B)
x2 + 3x  54
S
Name
Math 152B Final
Please do all of the following problems. All work and all answers must be done
on your own paper. Credit earned will be based on the steps that you show that
lead to the final solution. Good Luck!
Problem 1: Factor the following
A)
x5
Addition and Subtraction of Rational Expressions
Review of Addition and Subtraction of Fractions
Example 1
3
5
8
+
20
=
(2)(4)
=
20
20
2
=
(4)(5)
5
Notice the steps we have done to solve this problem. We first combined the numerators since the
denominator
Completing the Square and The Square Root Method
The Square Root Property
If we have
x2 = k
then we can take the square root of both sides to solve for x.
The Square Root Property
For any positive number k,
if
x2 = k
then
x =
or
x = 
Example
Solve
x2  6
Equations and Factoring
The Zero Product Theorems
Zero Product Theorem For Numbers
If a and b are numbers then
ab = 0
implies that
a = 0 or b = 0
We have a similar theorem for polynomials:
Zero Product Theorem For Functions
If f(x) and g(x) are functions
Problem Solving and Solving Roots
Work Problems
Example
Suppose that George can finish his homework in 5 hours and Carmen can finish her
homework in 4 hours. How many hours will it take for them to finish their homework
if they do it together?
Solution
We