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Module 3
Rational Algebraic Expressions
What this module is about
Rational expressions represent real numbers, the properties of real numbers can
be used to find the product and quotient of two rational expressions.
This module is about multiplication and division of rational expressions. As you
go over the exercises, you will develop skills in finding the product and quotient of the
given rational expressions. You will also recall some concepts on the laws of
exponents, multiplication and factoring of polynomials.
What you are expected to learn
This module is designed for you to:
1.
2.
3.
4.
5.
Recall some laws of exponents
Recall multiplication of polynomials
Recall factoring of polynomials
Multiply rational expressions
Divide rational expressions
How much do you know
1. Give the GCF of _2x – 4y _ .
x – 2y
2. Express #1 in simplest form.
a. 1
b. 2
c. x – 2
d. 2 – y
3. Factor the numerator of rational expression _18a2 + 27a + 10_.
3a + 2
4. Simplify the rational expression in #3.
a. a2 – 5
b. a(a – 5) c. 6a + 5
d. 6a + 2
1
5. Find the product of 2a2d  9b2c .
3bc
16ad2
6. Multiply
4a + 8
a2 – 25
a. __5___
5a + 25
and _a – 5 .
5a + 10
b. _a – 5_
a+5
c. __4___
5a + 25
d. _a – 4_
a+5
7. Find the measure of the area of the rectangle in simplest form.
2x + 4
x
__x3 – 4x___
x2 + 4x + 4
8. Divide:
__y_ _ ÷ __y__ .
y+3
y+5
a. _y + 5_
y+3
b. _y – 5_
y+5
c. _y + 3_
y–3
9. Find the quotient: (3m)2 ÷ 6m3 .
(2p)3
16p2
10. Divide
x – y ÷ ____x + y_ _ .
x2 – y2
x2 + 2xy + y2
2
d. _y – 3_
y+5
What you will do
Review 1
Multiplying and Polynomials
Let’s recall how to multiply polynomials using properties of exponents.
For Your Information
Humans grow until they reach physical maturity at about
18 years of age. After that most growth stops, although a
person’s hair and nails will continue to grow throughout his or her
life.
Some species of fish grow continually during their life span. Because of this,
some fish may be significantly larger than the average size for their species.
Think of a world in which humans continue to grow during their entire life span.
How do you imagine life on Earth would be different if humans continued to grow each
year of their lives?
Where w is the
weight, l is the
length, and a is a
constant for the
species.
The rate of growth of fish is
much greater when measure
by weight rather than by
length. We use w = al3 to
best relate the weight of a
fish to its length.
You may recall that in al3 where the base a and the base l are any real
numbers,1 is the exponent of a and 3 is the exponent of l. The exponent indicates how
many times the base is used as a factor.
al3 is an example of monomial.
Let’s recall the definition of monomials:
A monomial is a real number, a variable, or a product of real number and one or
more variables.
Based on the definition 5 is not a monomial because there is a variable in the
denominator.
y4
3
We can also say that 5 is not a polynomial because a polynomial is a monomial
y4
or a sum or difference of monomials.
Based on the definition of rational expression in the past modules 5 is a rational
expression.
y4
Let’s again define rational expressions:
A rational expression is an expression that can be written in the form p where p
and q are polynomials, q  0.
q
But you need to recall how to multiply polynomials because this same skill will
be used when you multiply rational expressions.
Before we multiply polynomials let’s review some properties of exponents for
multiplication.
Property of Exponents for Multiplication
1. For all real numbers a and all positive integers m and n:
am  an = am+n
In order to use the above property, the monomials must have the same base.
Example 1. Simplify: (2x2y5z)(3x3yz4)
(2x2y5z)(3x3yz4) = (2  3)(x2+3y5+1z1+4)
= 6x5y6z5
2. For all real numbers a and all positive integers m and n:
(am)n = amn
Example 2. Simplify: (2x6)5
(2x6)5 = (25)(x6 5) = 32x30
3. For all real numbers a and b and for all positive integers m:
(ab)m = ambm
Example 3. Simplify: (2xy)4
(2xy)4 = (24)(x4)(y4) = 16x4y4
4. For all real numbers a and b, b  0, and for all positive integers m:
a
b
m
= am
bm
4
Example 4. Simplify: 2y
x
2y
x
3
3
= 23y3 = 8y3
x3
x3
Multiplying Polynomials
Let’s begin by multiplying a monomial by a polynomial.
To multiply a polynomial by a monomial, you will often use the property of
exponents for multiplication: am  an = am+n
Example 1. Multiply –3x2 by x3 – 5x2 + 7x – 1
–3x2 (x3 – 5x2 + 7x – 1) = –3x5 +15x4 – 21x3 + 3x2
Now, let’s multiply two binomials. The distributive property can be used to
multiply two binomials (x + y)(a + b).
To multiply two binomials such as (x + y)(a + b), think of (a + b) as one factor.
Distribute by multiplying both x and y by (a + b). Then use the distributive property a
second time.
(x + y)(a + b) = x(a + b) + y(a + b)
= xa + ab + ya + yb
Try to relate these terms to the steps in the FOIL Method for multiplying two
binomials
To multiply two binomials, find:
F  the product of the two FIRST terms
O  the product of the two OUTSIDE terms
I  the product of the two INSIDE terms
L  the product of the two LAST terms
O
F
( x + y ) ( a + b)
I
L
( x + y ) ( a + b ) = xa + xb + ya + yb
F O
I
L
Example 2. Multiply (2a + b) by (a + 3b).
(2a + b)(a + 3b) = (2a)(a) + (2a)(3b) + (b)(a) + (b)(3b)
F
O
I
L
2
2
= 2a + 6ab + ab + 3b
= 2a2 + 7ab + 3b2
Combine like terms 6ab + ab
5
The FOIL method applies only to the multiplication of two binomials. When
multiplying longer polynomials, distribute the terms using either a horizontal or vertical
format.
Example 3. Multiply: (2y + y2 – 3y3)(4y – 5)
Horizontal Format:
(2y + y2 – 3y3)(4y – 5) = (2y + y2 – 3y3)(4y) + (2y + y2 – 3y3)(– 5)
= (8y2 + 4y3 – 12y4) + (–10y – 5y2 + 15y3)
= –10y + 3y2 + 19y3 – 12y4
Vertical Format: it is sometimes helpful to arrange the polynomials in descending order
before multiplying.
– 3y3 + y2 + 2y
4y – 5
15y3 – 5y2 –10y
–12y4 + 4y3 + 8y2
.
–12y4 +19y3 + 3y2 –10y
Multiplying Polynomials : Special Cases
Square of a Binomial: (a + b) 2 or (a – b) 2
 Square the first term
 Double the product of the two terms
a2
+2ab for (a + b) 2
–2ab for (a – b) 2
b2
 Square the last term
 Write the sum of the three term
(a + b) 2 = a2 + 2ab + b2
(a – b) 2 = a2 – 2ab + b2
Example 1. Simplify : (2a + 3b) 2
(2a + 3b) 2 = 4a2 + 12ab + 9b2
Example 2. Simplify : (3m2 – 11) 2
(3m2 – 11) 2 = 9m4 – 66m2 + 121
Product of the Sum and Difference of the Same Two Terms:
(a + b)(a – b)
 Square the first term
a2
 Square the last term
b2
 Write the difference of the two squares (a + b) (a – b) = a2 – b2
6
Example 3. (2r3 + 5) (2r3 – 5)
(2r3 + 5) (2r3 – 5) = 4r6 – 25
Practice Exercise:
Find the product of the following polynomials
1. ( a + 5)(a – 9)
2. ( h + 9)(h + 7)
3. ( 2m + 4)(m + 10)
4. ( 5x – 8)(2x – 3)
5. ( y + 4) 2
6. ( 2t – 2u) 2
7. ( h – 2q)(h + 2q)
8. ( 7m – 4)(7m + 4)
9. ( 5g – 3k)(5g + 3k)
10. ( 2 – 6y2)(2 + 6y2)
Factoring Polynomials
Do you look for products in packages that can be recycled? More and more
people do - because they are aware that “MAY PERA SA BASURA.’
When a product’s packaging is designed, many factors are considered: the
cost of materials, the attractiveness of its design and the convenience of its size. But
today, companies and consumers are increasingly concerned about what happens to a
package and how it affects the environment after being discarded. What factors will you
consider if you are asked to get involved in Waste Segregation Program?
TRIVIA TIME!
How long does trash last in a garbage dump?
In a recent study, newspaper in a 25-year old
bundle was readable. It also stated that glass bottles may
1 000 000 years to deteriorate.
take
In Algebra, what do we consider in factoring polynomials ?
First, we you must be able to give the common monomial factor of polynomials which is
actually the greatest common factor (GCF).
Example 1. Factor 8m4n2 + 18m3n2 – 6m2n
The GCF is 2m2n.
8m4n2 + 18m3n2 – 6m2n = 2m2n (4m2n) + 2m2n(9mn) – 2m2n(3)
= 2m2n ( 4m2n + 9mn – 3)
7
2m2n ( 4m2n + 9mn – 3) is the factored form of 8m4n2 + 18m3n2 – 6m2n.
How to factor polynomial in a form x2 + bx + c, where the sign of the last term is
positive?
Factoring a trinomial of the form x2 + bx + c means to express the trinomial as the
product of two binomials of the form (x + a)(x + b).
Example 2. Factor x2 + 7x + 10.
(x + a)(x + b) = x2 + bx + ax + ab
7x
10
The product of the second terms a and b of binomials must be 10 and the sum of
the outer and inner term, bx and ax, must be 7x.
Factors of 10
Possible Binomial Factors
1, 10
-1, -10
2, 5
-2, -5
(x + 1)(x + 10)
(x –1)(x –10)
(x + 2)(x + 5)
(x –2)(x – 5)
Sum of Outer and Inner
Products
1x + 10x = 11x
-1x – 10x = -11x
2x + 5x = 7x 
-2x – 5x = -7x
So, x2 + 7x + 10 = (x + 2)(x + 5).
Example 3. Factor m2 – 5m + 6.
Factors of 6
Possible Binomial Factors
1, 6
-1, -6
2, 3
-2, -3
(m + 1)(m + 6)
(m –1)(m –6)
(m + 2)(m + 3)
(m –2)(m – 3)
Sum of Outer and Inner
Products
1m + 6m = 7m
-1m – 6m = -7m
2m + 3m = 5m
-2m – 3m = -5m 
So, m2 – 5m + 6= (m –2)(m – 3).
Example 4. Factor a2 – 7ab + 12b2.
Factors of
12b2
1b, 12b
2b, 6b
3b, 4b
Possible Binomial Factors
(a + 1b)(a + 12b)
(a + 2b)(a + 6b)
(a + 3b)(a + 4b)
So a2 – 7ab + 12b2 = (a + 3b)(a + 4b)
8
Sum of Outer and Inner
Products
12ab + 1ab = 13ab
6ab + 2ab = 8ab
4ab + 3ab = 7ab 
Factoring a trinomial of the form x2 + bx + c, where the sign of the last term is
negative.
Lucy made up this riddle for Jackie “ “ I’m thinking of two numbers. The product
of the numbers is -24, and the sum of the numbers is -2.”
Jackie tried a few combinations in her head before coming up with the answer: “
The numbers must be -6 and +4.” Is any other combination possible?
Example 5. Factor x2 + 3x – 10.
Factors of 10
Possible Binomial Factors
1, –10
–1, 10
2, –5
–2, 5
(x + 1)(x – 10)
(x –1)(x +10)
(x + 2)(x – 5)
(x –2)(x + 5)
Sum of Outer and Inner
Products
1x – 10x = –9x
–1x +10x = 9x
2x – 5x = –3x
–2x + 5x = 3x 
So, x2 + 3x – 10 = (x –2)(x + 5).
Example 6. Factor m2 – 5mn – 14n2.
Factors of
– 4n2
1n, –14n
–1n, 14n
2n, –7n
–2n, 7n
Possible Binomial Factors
(m +1n)(m –14n)
(m –1n)(m +14n)
(m +2n)(m –7n)
(m –2n)(m + 7n)
Sum of Outer and Inner
Products
14mn –1mn =13mn
–14mn +1mn = –13mn
–7mn + 2mn = –5mn 
7mn – 2mn = 5mn
So, m2 – 5mn – 14n2= (m +2n)(m –7n).
Factoring ax2 + bx + c.
When the coefficient of the first term of a trinomial is not 1, the number of
possible binomial factors increases.
Example 7. Factor 2x2 – 3x – 5.
Factors of
2x2
2x, x
Factors of
–5
–1, 5
1, –5
Possible Binomial
Factors
(2x – 1)(x + 5)
(2x + 5)(x – 1)
(2x +1)(x – 5)
(2x – 5)(x + 1)
So, 2x2 – 3x – 5 = (2x – 5)(x + 1).
9
Sum of Outer and Inner
Products
10x –1x = 9x
–2x + 5x = 3x
–10x +1x = –9x
2x – 5x = –3x 
Example 8. Factor 6x2 + 19x + 3.
Factors of
6x2
3x, 2x
Factors of
3
3,1
Possible Binomial
Factors
(3x + 3)(2x + 1)
(3x + 1)(2x + 3)
6x, x
3,1
(6x + 3)(x + 1)
(6x + 1)(x + 3)
2
So, 6x + 19x + 3 = (6x + 1)(x + 3).
Sum of Outer and Inner
Products
3x + 6x = 9x
9x + 2x = 11x
6x + 3x = 9x
18x + x = 19x 
Example 9. Factor 4x2 + 59xy –15y2.
Factors of
4x2
4x, x
Factors of
–15y2
3y, –5y
–3y, 5y
1y, –15y
–1y, 15y
Possible Binomial
Factors
(4x + 3y)(x – 5y)
(4x – 3y)(x + 5y)
(4x + y)(x –15y)
(4x – y)(x +15y)
So, 4x2 + 59xy –15y2= (4x – y)(x +15y).
Practice Exercise
Factor the following polynomials:
.
1. 2m3y – 12m2y4
2. 33w3y2 + 11w2y2
3. 9cd4 + 6c2d2 – 3c3d
4. r2 + 6r + 8
5. c2 + 20c + 91
6. n4 – 8n2 + 12
7. x2 + 4x – 5
8. b2 – b – 56
9. k2 – 13k – 30
10. w2 – 2wz – 8z2
11. p2 – 11px – 80x2
12. 3x2 – 22x – 16
13. 2m2 + 7m + 5
14. 2b2 + 9b – 11
15. 4c2 – 4c – 3
10
Sum of Outer and Inner
Products
–20xy + 3xy = –17xy
20xy – 3xy = 17xy
–60xy + xy = – 59xy
60xy – xy = 59xy 
Factoring: Special Cases
When you square a binomial, the product is called a perfect square trinomial.
(x + 4) 2 = (x + 4)(x + 4) = x2 + 8x + 16
square of the 1st term
twice the product
of the two terms
square of the last term
In general, polynomials that are perfect square trinomials factor as follows:
x2 + 2xy + y2 = (x + y)(x + y) = (x + y) 2
x2 – 2xy + y2 = (x – y)(x – y) = (x – y) 2
Example 10. Factor 36p4 + 36p2y + 9y2.
You must first identify if the polynomial is a perfect square trinomial before you
apply the rule.
36p4 + 36p2y + 9y2 = (6p2) 2 + 2(6p2)(3y) + (3y) 2
= (6p2 + 3y)(6p2 + 3y) or (6p2 + 3y) 2
When you multiply binomials that are the sum and a difference of the same two
numbers, the product is a difference of two squares.
( x + 9)(x – 9) = (x) 2 – (9) 2 = x2 – 81
square of the 1st term
square of the last term
x2 – 81 is a difference of two squares.
You are now ready to multiply rational expressions.
Lesson 1
Multiplying Rational Expressions
When you multiply rational numbers, you look for common factors to simplify the
product or to simplify the multiplication.
Examples: Find the product of 2 and 9
3
10
1
3
2 9 =2 9= 3
3 10 3  10 5
1
5
1
or
3
2 9 =2  9= 3
3 10 10 3 5
5
11
1
Simplify: 15a4b
3a3
5
15a4b = 15a4b = 5b
3a3
3a5
a
a
The rule for multiplying rational expressions is similar to the rule for multiplying
rational numbers.
Multiplication of Rational Expression
If P and R are rational expression with Q  0 and S  0, then P  R = P  R or PR
Q
S
Q S Q S
QS
To make your work simpler, you may assume that all denominators in the
following exercises do not equal zero.
Example 1. Find the product of m + y and
m2 _
2m
(m + y) 2
m+y 
m2 _ = (m + y)m2_ = (m + y) m  m_ = __m___
2m
(m + y) 2
2m(m + y) 2 2m(m + y) (m + y) 2(m + y)
Take note that we factor m2 and (m + y) 2 and use the fundamental property of
rational expressions to write the product in lowest terms.
Fundamental Property of Rational Expressions states that
If _P_ is a rational expression and if K represents any rational expression, with
Q
K  0, then PK = _P .
QK Q
This property is based on the identity property of multiplication, since
PK = _P  K = P  1 = P
QK Q
K Q
Q
Example 2. Find the product of _x + 5_ and ___12x2 ____
3x
x 2 + 7x + 10
_x + 5_  ___12x2 ____ = _x + 5_  __4x _3x___
3x
x 2 + 7x + 10
3x
(x + 5) (x + 2)
= __4x__
x+2
12
Example 3. Find the product of _m2 + 3m__ and m2 – 5m_+ 4
m2 – 3m – 4
m 2 + 2m – 3
First factor the numerators and denominators whenever possible. Then write the
product in lowest terms.
_m2 + 3m__  m2 – 5m_+ 4 = __m (m + 3)__  (m – 4)(m – 1)_
m2 – 3m – 4
m 2 + 2m – 3
(m – 4)(m + 1) (m + 3) ( m – 1)
= __m__
m+1
Example 4. Find the product of __x2 – x – 6 _ and x2 + 7x + 12_
9 – x2
x2 + 4x + 4
__x2 – x – 6 _  x2 + 7x + 12_ = _(x – 3)(x + 2)_  (x + 3)(x + 4)
9 – x2
x2 + 4x + 4
– (x – 3)(x + 3) (x + 2)(x + 2)
Note that 9 – x2 = (3 – x)(3 + x) or – (x – 3)(x + 3)
__x2 – x – 6 _  x2 + 7x + 12_ = __x + 4_ = – _x + 4_
9 – x2
x2 + 4x + 4
– (x + 2)
x+2
Try this out
A. Find each product. Assume that no denominator is equal to zero.
1. _a4b_  _c_
b3c
a3
6. _5n – 5
3
2. 10n3  12n3x9
6x7
25n5x5
7. – ( 2a + 7c)
6
3.
_2a_
b
3
 _5b3
16a
4. _8m_ _3_
m2
2c
2
5. _7x3y5_  _44z5_
11z2
21x7y2
8. _a2 – b2
4
 __9__
n–1
 ___36__
–7c – 2a
 __16__
a–b
9. _ 3
 _(x – y) 2
x–y
6
10. m2 – n 2  _7mn_
m–n
m+n
13
B. Multiply. Assume that no denominator is equal to zero.
1. _x2 – 16_  _x + 4_
8x
x–4
6. _ b + a
b–a
2. _a2 – b2_  _a – 1_
a2 – 1
a–b
7. 3mn2 – 3m_  __3m__
n
n2 –1
3. _3k + 9_  __k2__
k
k2 – 9
8. _ ___x__ _  _2x + 10_
x2 + 8x + 15
x2
4. _3m – 6 _  _m + 3__
m2 – 9
m2 – 2m
9. _ x – 5 _  __x – 2__
x2 – 7x + 10
3
5. _x2 – y2_  __x__
y
x–y
 _a2 – b2_
2a
10. _ b2 + 20 b + 99 
b+9
____b + 7___
b2 + 12b + 11
C. Find the measure of the area of the rectangle in simplest form
1.
_x + 7_
x2 – 25
x2 + 10x + 25
x2 – 49
2.
2x + 3
x2
2x + 3
x2
3. x2 – 15x + 50  x2 – 11x + 24
x2 – 9x + 20
x2 – 18x + 80
4. y2 + 3y3  ___2y + y2 __
y2 – 4
y + 4y2 + 3y3
14
Lesson 2
Dividing Rational Expressions
You should recall that two numbers whose product is 1 are called multiplicative
inverses or reciprocals.
To find the quotient of two fractions, you multiply by the reciprocal of the divisor.
2÷3=2 4=8
3 4 3 3 9
You can use the same method to divide rational expressions.
Division of Rational Expression
If P and R are rational expression with Q  0 and S  0, then P ÷ R = P  S or PS
Q
S
Q S
Q R
QR
Example 1. Divide: 5_ ÷ _y_
x
z
_5_ ÷ _y_ = _5_  _z_ = _5z_
x
z
x
y
xy
Example 2. Divide: _5__ ÷ _ z__
z+3
z–6
_5__ ÷ _ z__ = __5__  _z – 6 _
z+3
z–6
z+3
z
= _5 (z – 6 )_= __5z – 30 _
z (z + 3)
z2 + 3z
Example 3.
(5y)2_ ÷ _20y3_
(2x)3
32x2
(5y)2_ ÷ _20y3_ = (5y)2_  _32x2 _
(2x)3
32x2
(2x)3
20y3
5
4 1
= 25y2_  _32x2 = 5_
8x3
20y3
xy
x
4
y
15
Example 4. Divide _2x__ by (x – 1).
x +1
_2x__ ÷ (x – 1) = _2x__  _ 1__
x +1
x +1
x–1
= ____2x__ __
(x +1)(x – 1)
= __2x __
x2 – 1
Example 5.
___x2 – 4___ ÷ _(x + 2)(x + 3)_
(x – 2 )(x + 3)
2x
___x2 – 4___ ÷ _(x + 2)(x + 3)_ = ___x2 – 4___  _
2x_____
(x – 2 )(x + 3)
2x
(x – 2 )(x + 3) (x + 2 )(x + 3)
Be sure that all numerators and denominators are factored.
= _(x + 2)(x – 2)  _
2x_____
(x – 2 )(x + 3) (x + 2 )(x + 3)
= ___2x___
(x + 3) 2
Example 6.
_m2 – 4_ ÷ _2m + 4m_
m2 – 1
1–m
_m2 – 4_ ÷ _2m + 4m_ = _m2 – 4_  __1 – m__
m2 – 1
1–m
m2 – 1
2m + 4m
= _(m + 2)(m – 2)_  __1 – m__
(m + 1)(m – 1)
2m(m + 2)
since 1 – m = – 1, then
m–1
1
-1
= _(m + 2)(m – 2)_  __1 – m__
(m + 1)(m – 1)
2m(m + 2)
1
= _–1(m – 2)_ = __2 – m__
2m(m + 1)
2m(m + 1)
16
Try this out
Answer the following:
A. Divide and express your answer in lowest term if necessary.
__6s2__
2s2 + 4s
1. 3 ÷ _2_
p
p
6. _ 3s
s+2
÷
2. _3r2_ ÷ _8r4_
9r3
6r5
7. _a + b
2
÷ __(a+ b)2_
(2r)2
3. _25m10 ÷ _15m6_
9m5
20m4
8. _ 5m + 25
10
4. _12n2_ ÷ _9n3_
(4n)3
32n4
9. _ 2 – y
8
5. –6x4 ÷ _(2x2)2_
3x5
–4
10. 2r + 2p ÷
8z
÷ _6m + 30_
12
÷ _y – 2 _
12y
_r2 + rp_
72
B. Divide and express your answer in lowest term if necessary.
1. _ _ 3 _
÷ __– 12 _
b2 – 5b + 6
b2 – b – 2
2.
4y + 12_ ÷ __y2 – 9 __
2y – 10
y2 – y – 20
3. _9(y – 4)2_ ÷ _3(y – 4)_
8(z + 3)2
16(z + 3)
4. _ x2 – 16
x+3
5. _ m2 – 16
4–m
6.
÷ _x – 4_
x2 – 9
÷_– 4 – m _
–4+m
_6r – 18_ ÷ _4r – 12_
3r2 + 2r – 8
12r –16
7. _k2 – k – 6 _ ÷ _k2 + 2k – 3 _
k2 + k – 12
k2 + 3k – 4
8. _y2 + y – 2 ÷ _y + 2_
y2 + 3y – 4
y+3
17
9. _m2 + 3m + 2 _ ÷ _m2 + 5m + 6_
m2 + 5m + 4
m2 + 10m + 24
10. _n2 + 2np – 3p2_ ÷ _n2 + 4np + 3p2_
n2 – 3mp + 2p2
n2 + 2np – 8p2
C. Divide and write your answer in lowest term.
1. _2k2 + 3k – 2 _ ÷ _k2 + k – 2 _
6k2 – 7k + 2
4k2 – 5k +1
2. _2m2 – 5m – 12 _ ÷ __4m2 – 9_ _
m2 – 10m + 24
m2 – 9m +18
3. _r2 + rs – 12s2_ ÷ _r2 – 2rs – 3s2_
r2 – rs – 20s2
r2 + rs – 30s2
4. _(x + 1)3 (x + 4)_ ÷ _x2 + 2x + 1 _
x2 + 5x + 4
x2 + 3x + 2
5. _(q – 3)4 (q + 2)_ ÷ _q2 – 6q + 9 _
q2 + 3q + 2
q2 + 4q + 4
Let’s Summarize
Rational expression is the quotient of two polynomials with denominator not
equal to zero.
The fundamental property of rational expressions permits us to write a
rational expression in lowest terms, in which numerator and denominator have no
common factor other than 1.
Fundamental Property of Rational Expressions states that
If _P_ is a rational expression and if K represents any rational expression, with
Q
K  0, then PK = _P .
QK Q
18
This property is based on the identity property of multiplication, since
PK = _P  K = P  1 = P
QK Q
K Q
Q
Multiplying Rational Expressions:
If P and R are rational expression with Q  0 and S  0, then P  R = P  R or PR
Q
S
Q S Q S
QS
Dividing Rational Expressions:
If P and R are rational expression with Q  0 and S  0, then P ÷ R = P  S or PS
Q
S
Q S
Q R
QR
19
What have you learned
Perform the given operation. Write your answer in lowest term. Assume that no
denominator is equal to zero.
1. In the rational expressions ____x + y___ , give the factors of the
denominator.
x2 + 3xy + 2y2
a. (x – y)(x + 2y)
c. (x + y)(x + 2y)
b. (x + y)(x – 2y)
d. (x – y)(x – 2y)
2. The simplified form of #1 is
a. __1___
x+y
b. __1___
x + 2y
c. __1___
x–y
d. __1___
x – 2y
3. Simplify _z2 – 3z_ .
z–3
4. Find the product of 7  _a2 .
9
b
5. Find the product of 5x2y  _12a2b_ .
8ab
25x
6. Multiply
x2 + x – 12
x+2
7. Multiply
b2 + 19b + 84
b–3
and __x + 4__.
x2 – x – 6
and _ __b2 – 9___ .
b2 + 15b + 36
8. Find the quotient of p4 ÷ _–(p2) 3_ .
2q
4q
9. Divide:
7a2b __ ÷ _ ___ 3a ___ .
x2 + x – 30
x2 + 15x + 54
10. Divide:
m2 + 4m – 21 ÷ _____m2 – 9___ .
m2 + 8m + 15
m2 + 12m + 35
20
Answer Key
How much do you know
x – 2y
b. 2
18a2 + 27a + 10 = (3a + 2)(6a + 5)
c. 6a + 5
3ab
8d
6. c. __4___
5a + 25
1.
2.
3.
4.
5.
7. 2x + 4  __x3 – 4x___ = 2(x + 2)  __x (x + 2)(x – 2)__
x
x2 + 4x + 4
x
(x + 2)(x + 2)
= 2(x – 2) square units
8. a. _y + 5_
y+3
9. 9m2  16p2 = _3_
8p3
6m3 mp
10.
___x – y__  _(x + y)(x + y) = 1
(x – y)(x + y)
x+y
Review 1
Practice Exercises
1. a2 – 4a – 45
2. h2 + 16h + 63
3. 2m2 + 24m + 40
4. 10x2 – 31x + 24
5. y2 + 8y + 16
6. 4t2 – 8tu + 4u2
7. h2 – 4q2
8. 49m2 – 16
9. 25g2 – 9k2
10. 4 – 36y4
21
Review 2
Practice Exercises
1. 2m2y ( m – 6y3)
2. 11w2y2 (3w + 1)
3. 3cd ( 3d3 + 2cd – c2)
4. (r + 2)(r + 4)
5. (c + 13)(c + 7)
6. (n2 – 2)(n2 – 6)
7. (x – 1)(x + 5)
8. (b – 8)(b + 7)
9. (k + 2)(k – 15)
10. (w + 4z)(w – 2z)
11. (p + 5x)(p – 16x)
12. (3x + 2)(x – 8)
13. (2m + 5)(2m + 1)
14. (2b + 11)(b – 1)
15. (2c – 3)(2c + 1)
Lesson 1
Try this out
1. _a_
b2
2. 4n
5x3
6. 15
3. _5a2
2
4. _18_
mc2
8. 4(a + b)
5. _4y3z3_
3x4
10. 7mn
7. 6
9. x – y
2
B.
1. _(x + 4)(x – 4)(x + 4)_ = (x + 4) 2
8x ( x – 4)
8x
22
2. _(a – b)(a+b)(a – 1)_ = a + b
(a – b)(a +1)(a – 1)
a +1
3. _3(k + 3)_  ____k2_____ = _3k_
k
(k + 3)(k – 3) k – 3
4. ____3(m – 2)(m + 3)____ = ___3___
m(m – 2)(m + 3)(m – 3) m(m – 3)
5. _x(x + y)(x – y)_ = _x(x + y)_
y (x – y)
y
6. _(b + a)(b + a)(a – b)_ = (b + a) 2 or – (b + a) 2
– 2a(a – b)
– 2a
2a
Take note that (b – a) = – (a – b)
7. 3m(n – 1)(n + 1)(3m)_ = 9m2
n(n – 1)(n + 1)
n
8. ___2x (x + 5)___ = ___2__
x2 (x + 3)(x + 5) x(x + 3)
9. _(x – 5)(x – 2)__ = _1_
3(x – 5)(x – 2)
3
10. _(b + 11)(b + 9)(b + 7)_ = _b + 7_
(b + 11)(b + 9)(b + 1)
b+1
C.
1. _
(x + 7)(x + 5)(x + 5)
= ___x + 5____
(x + 7)(x – 7)(x + 5)(x – 5)
(x – 7)(x – 5)
2. _(2x + 3) 2_
x4
3. _(x – 5)(x – 10)(x – 8)(x – 3)_ = x – 3
(x – 5)(x – 4)(x – 8)(x – 10)
x–4
4. _y2(1 + 3y)___  __y(2 + y)____
(y – 2)(y + 2)
y(1 + 4y + 3y2)
____y3(1 + 3y)(2 + y)_______ = ____y2_____
y (y – 2)(y + 2)(3y + 1)(y + 1)
(y – 2)(y + 1)
23
A.
1. 3_
2
6. _ 3s  _2s(s + 2)_ = 1
s+2
6s2
2. _1__
4
7. _a + b
2
3. _100m3_
27
8. _5(m + 5)  __12____ = 1
10
6(m + 5)
4. _2_
3
9. Note that 2 – y = – (y – 2)
Ans. –3y
2
10. 18_
rz
5.
2_
x5
B.
1. _b + 1_
-4(b -3)
2.
 _ 4r2__ = _ 2r2__
(a+ b)2
a+b
6. __6__
r+2
2(y + 4)_
y–3
7. _k + 2_
k+3
3. _6(y – 4)_
z+3
8. _y + 3_
y+4
4. (x + 4)(x – 3)
9. _m + 2_
m+1
5. m – 4
10. _n + 4p_
n+p
C.
1. _(2k – 1)(k + 2)(4k – 1)(k – 1) = _4k – 1_
(2k – 1)(3k – 2)(k + 2)(k – 1)
3k – 2
2. _(2m + 3)(m – 4)(m + 6)(m – 3)_ = _m – 3_
(m – 6)(m – 4)(2m – 3)(2m + 3)
2m – 3
3. _(r + 4s)(r – 3s)(r – 5s)(r + 6s)_ = _r + 6s_
(r – 5s)(r + 4s)(r – 3s)(r + 5)
r+s
4. _(x + 1) (x + 4)(x + 2)(x + 1)_ = (x + 2)(x + 1)
(x + 4)(x + 1)(x + 1)2
24
5. _(q – 3)4 (q + 2)(q + 2)(q + 2)_ = _(q – 3)2 (q + 2)2 _
(q + 2)(q + 1)(q – 3)(q – 3)
q+1
What have you learned
1. c
2. b
3. z
4. _7a2
9b
5. 3axy
10
6. (x + 4)2 (x – 3) = (x + 4)2
(x + 2)2 (x – 3)
(x + 2)2
7. (b + 12)(b + 7)(b – 3)(b + 3) = b + 7
(b – 3)(b + 12)(b + 3)
8. __2_ or – _2_
– p2
p2
9. _7a2 b(x + 9)(x + 6) = 7ab(x + 9)
3a (x + 6) (x – 5)
3(x – 5)
10. _(m + 7)2 (m – 3)(m + 5) = (m + 7)2
(m + 3)2 (m – 3)(m + 5)
(m + 3)2
25