CHILDREN’S GRACE OF MARY TUTORIAL AND LEARNING CENTER, INC.
DUMADAG SUBDIVISION, NEW CARMEN, TACURONG CITY
CONTACT NUMBER: 0977-804-5567
MATHEMATICS
Module 4
CHILDREN’S GRACE OF MARY TUTORIAL AND LEARNING CENTER, INC.
DUMADAG SUBDIVISION, NEW CARMEN, TACURONG CITY
CONTACT NUMBER: 0977-804-5567
MATHEMATICS
Name: _______________________________________________________ Score: __________
MODULE: 4
LESSON 1: Multiplying and Dividing Rational Expressions
Lesson Objectives
At the end of this lesson, the student will be able to multiply and divide rational expression.
What I know.
Fraction Prime Factorization of the Cancel Pairs of Common Product
Numerators and Denominators Factors
1. 8 7 2. 2 ∙2 7 2. 2 ∙2 7 1
∙ ∙ ∙
14 12 2 ∙7 2 ∙2 ∙ 3 2 ∙7 2 ∙2 ∙ 3 3
2. 10 12
∙
24 15
3. 24 12
∙
54 9
4. 16 30
∙
24 8
5. 4 45
∙
35 12
What you need to know.
Rule in Multiplying Rational Expressions:
P R PR
∙ = , where P, Q, R, and S are polynomials in one variable and Q ≠ 0, R ≠ 0, and S ≠ 0.
Q S QS
Multiplying Rational Expression Procedure:
1. Write each numerator and denominator in factored form.
2. Divide out any numerator factor with any matching denominator factor.
3. Multiply the numerators and also the denominators.
4. Simplify, if possible.
Examples:
a5 5 5 a5
a. ∙ = Multiply the numerators and denominators.
10 a3 10 a3
5 a3 ∙ a3
= Factor the numerator and denominator.
5 a3∙ 2
5 a3 ∙ a3
= Write a product, one factor containing the GCF of the numerator and
5 a3∙ 2
denominator, and the other containing the remaining factors.
2
a
= 1∙
2
a2
= (answer)
2
2
30 b2 4 c 120 b2 c2
b. ∙ = Multiply.
6 c 15 b4 90 b4 c
30 b3 c ∙ 4 c
= Factor the numerator and denominator.
30 b3 c ∙ 3 b2
30 b3 c ∙ 4 c
= Write a product, one factor containing the GCF of the numerator and
30 b3 c ∙ 3 b2
denominator, and the other containing the remaining factors.
4c
=1 ∙ 2
3b
4c
= (answer)
3 b2
Please see page 67-68 for some example.
Rule in Dividing Rational Expressions:
P R P S
= ∙ , where P, Q, R, and S are polynomials in one variable and Q ≠ 0, R ≠ 0, and S ≠ 0.
Q S Q R
Dividing Rational Expression Procedure:
1. Write the equivalent multiplication statement using the reciprocal of the divisor.
2. Factor the numerator and denominator.
3. Divide out any numerator factor with any matching denominator factor.
4. Multiply the numerators and denominators.
5. Simplify, if possible.
Examples:
7 7 7 12 x
a. = ∙ Write the equivalent multiplication expression.
6 x 12 x 6 x 7
7 6 x ∙2
= ∙ Factor and apply cancellation.
6x 7
= 2 (answer) Multiply.
a5 2 a5 1
b. 4a = . ∙ Write the equivalent multiplication expression.
4 4 4 a2
a3
= (answer) Quotient Rule for Exponents
16
(answer)
What I have learned.
Multiply and simplify each.
7 x5 2 x 5 10x²
1. ∙ 3.
x 4 21 7
d 40 25−x 2 ∙ x−5
2. ∙ 4. 2
12 d x2 +3 x x −9
What I can bring home.
Find the area of the following. (See example on page 71)
3x 4 x²
1. 2. 3 x +3
7
x ²−1
8x
LESSON 2: Adding and Subtracting Rational Expressions
Lesson Objective
At the end of this lesson, the student will be able to:
a. add and subtract rational algebraic expressions; and
b. solve problems involving rational algebraic expressions.
What I know.
Add the fractions. Write your answer in lowest term.
Fractions LCD Fractions with a
Common Denominator Sum
5 3 10 3 13
1. +8 8 + 8
4 8 8
1 4
2. +
3 5
5 3
3. +
6 15
4 5
4. +
18 12
What you need to know.
Addition/ Subtraction of Rational Expression Rules:
A B A B A +B A B A−B
If and are any two rational expressions, then + = and - = , where C ≠ 0.
C C C C C C C C
Procedure in Adding/ Subtracting Rational Expressions with Like Denominator
1. Add (or subtract) the numerators.
2. Retain the common denominator.
3. Simplify the result.
Examples:
2a 3 2 a+3
a. + = Add the numerators and keep the same denominator.
4b 4b 4b
8 d−3 4 d+12 8 d−3+ 4 d +12
b. + = Add the numerators and keep the same denominator.
9 9 9
12d +9
= Combine like terms.
9
3(4 d +3)
= Factor the numerator
9
³
(4 d +3)
= Divide out common factor 3.
3
(See some examples on page 78-79.)
Procedure in Adding/ Subtracting Rational Expressions with Different Denominators
1. Find the least common denominator (LCD).
2. Write the equivalent expression of each rational expression.
3. Add or subtract the numerators and keep the LCD.
4. Simplify the result, if possible.
Procedure in Finding the LCD
1. List the different denominators that appear in the rational expression.
2. Factor each denominator completely.
3. For each unique factor, compare the number of times it appears in each factorization. Write a factored form that
includes each factor the greatest number of times it appears in the denominator factorizations.
Examples:
5 3
1. Find the LCD of and .
4 ab 10 bc
Solution: Factor each denominator.
4ab = 2²ab ; 10bc = 2 ∙5bc
The LCD is 2² ∙5abc or 20abc. Convert each expression with 20abc as the denominator. Multiply the
numerator and denominator of the first rational expression by 5c and the second by 2a.
5 5 ∙5 c 25 c
= =
4 ab 4 ab∙ 5 c 20 abc
3 3∙2a 6a
=
10 bc 10 bc ∙ 2a
= 20 abc
2. Add or subtract.
What I have learned.
Add or subtract:
a 5 5 7
1. + 2. . –
a+3 a a ²+5 a 3 a+15
What I can bring home.
Answer page 83, Practice and Application # 1-5.
REVIEWER:
I. TRUE or FALSE. Write True if the statement is correct and False if it is wrong on the blank.
________1. Factoring is finding two or more factors of a number or a polynomial.
________2. The GCF of 6a and 18ab is 6a.
________3. A difference of two squares has a middle term.
________4. When x + 3 is multiplied by x – 3, the product is a perfect square trinomial.
II. Fill in each box to complete each equation.
1. x² + 19x + 12 = (4x + 3) (x + 4) 3. x² - 9x – 9 = ( x + 3) ( x - 3)
2. x² + 11x – 2 = ( x – 1) (x + 2) 4. x² - 17x – 5 = (3x – 5) ( x + 1)
III. Factor each expression completely.
1. 25 + 45x 2. 24a + 48b 3. 9a² - 49 4. a³ + 27
IV. Simplify each rational expression.
40 45 a ² b 3 x −2
1. 2. 3.
60 30 ab 2−3 x
V. Perform the indicated operation.
7 x5 2 x5 3x 4
1. 4 ∙ 2. 10x² 3. +
x 21 7 5y 5y
VI. Problem Solving. Show your solution. (5 pts each)
4x
1. What is the area of a square if the length of each side is ?
5
5 x2
2. Evaluate when x = 4 and y = 10.
6y
