In-class Handout, More 6.6 and Section 7.1, July 12th, 2012
Section 6.6 (More Examples)
(Similar to Q24 State Exam)
Solve for 𝑥:
𝑥 2 − 10𝑥 − 24 = 0
Solutions: 𝑥 = 12, −2
(Similar to Q25 State Exam)
Solve for 𝑥:
4𝑥 2 − 4𝑥 − 15 = 0
5
2
Solutions: 𝑥 = , −
3
2
(Similar to a difficult MyMathLab problem)
Solve for 𝑥:
20𝑥 4 = 90𝑥 3 + 50𝑥 2
1
Solutions: 𝑥 = 0, 5, − 2
Chapter 1 Selected Topics Review
Simplifying Fractions (1.2, page 12):
Prime factor and simplify the fraction:

42
=
68
2∙3∙7
2∙2∙17
=
2∙3∙7
2∙2∙17
=
3∙7
2∙17
=
21
34
Once factored, cancel any factors common to the numerator and denominator
Division by zero (1.6, page 53):
0
8


=0
5
0
0
= Undefined
0
= Undefined
Zero divided by any non-zero number is zero
Any number divided by zero is undefined
Section 7.1 Simplifying Rational Expressions
Definition of a Rational Expression: A rational expression is a fraction with a polynomial in the numerator and a
polynomial in the denominator.
Examples of Rational Expressions:
7
𝑥
𝑥 2 −1
𝑥 2 −3𝑥−4
𝑥 3 −𝑥 2 +𝑥−1
14
5
𝑥+1
𝑥 2 −𝑥−2
𝑥−1
Recalling that division by zero is undefined, it is possible for rational expressions to be undefined for some values of
the variables.
7
𝑥
𝑥 2 −1
𝑥 2 −3𝑥−4
𝑥 3 −𝑥 2 +𝑥−1
14
5
𝑥+1
𝑥 2 −𝑥−2
𝑥−1
In general, rational expressions are defined for all values of 𝑥 (i.e. all real numbers) as long as the denominator does
not simplify to zero for a given value of 𝑥. To find values of 𝑥 for which the rational expression is not defined, set the
denominator equal to zero and solve the equation.

For what values of 𝑥 are
𝑥
5
,
𝑥 2 −1
𝑥+1
𝑥 2 −3𝑥−4
,
𝑥 2 −𝑥−2
, &
𝑥 3 −𝑥 2 +𝑥−1
𝑥−1
defined?
Simplifying Rational Expressions
1. Factor the numerator and denominator completely
2. Cancel any common factors
3. You do not need to multiply/FOIL any results. The factored form is taken by MyMathLab and generally
preferred to the multiplied/FOILed/expanded form.
Examples:
Difference of Squares
Simplify
𝑥 2 −1
𝑥+1
=
(𝑥−1)(𝑥+1)
𝑥+1
=
Simplified Equivalents
(𝑥−1)(𝑥+1)
𝑥+1
=𝑥−1
For all values of 𝑥 except 𝑥 = −1
Factoring Trinomials (𝑎 = 1)
Simplify
𝑥 2 −3𝑥−4
𝑥 2 −𝑥−2
(𝑥−4)(𝑥+1)
(𝑥−4)(𝑥+1)
= (𝑥−2)(𝑥+1) = (𝑥−2)(𝑥+1) =
𝑥−4
For all values of 𝑥 except 𝑥 = 2, −1
𝑥−2
Factor By Grouping
Simplify
𝑥 3 −𝑥 2 +𝑥−1
𝑥−1
=
(𝑥 2 +1)(𝑥−1)
=
𝑥−1
(𝑥 2 +1)(𝑥−1)
𝑥−1
= 𝑥2 + 1
For all values of 𝑥 except 𝑥 = 1
Example MyMathLab Problems, Simplify:
1−𝑧
𝑦 2 −5𝑦
𝑛2 −2𝑛
𝑥+2
𝑧−1
5𝑦−𝑦 2
𝑛2 −9𝑛
𝑥 2 −7𝑥−18
−
𝑎−10
−5𝑥−1
10−𝑎
20𝑥+4
Example Q23 State Exam, Simplify:
𝑥 2 −4𝑥+4
2𝑥 2 +7𝑥−4
3𝑥 2 +7𝑥+4
4𝑥 2 +4𝑥−3
𝑥 2 −5𝑥+6
4𝑥 2 −1
𝑥 2 +6𝑥+5
4𝑥 2 −4𝑥−15
Answers of MyMathLab Examples:
−1
−1
𝑛−2
1
𝑛−9
𝑥−9
−
+1
Answers of Q23 State Exam Examples:
𝑥−2
𝑥+4
(3𝑥+4)
2𝑥−1
𝑥−3
2𝑥+1
(𝑥+5)
2𝑥−5
1
4