1. Section 9.4
Rational Expressions

2. Rational Expressions Factor.
ALGEBRA 2 LESSON 9-4
(For help, go to Lesson 5-4 and Skills Handbook page 843.)
Factor.
1. 2x2 – 3x + 1    2. 4x2 – 9    3. 5x2 + 6x + 1    4. 10x2 – 10
Multiply or divide.
5. •      6.  •       •       •
9. ÷ 4     10.  ÷     11.  ÷      12.  ÷ 3 8 5 6 1 2 4 16 7 9 15
Check Skills You’ll Need
9-4

3. Check: (2x – 1)(x – 1) = 2x2 – 2x – 1x + 1 = 2x2 – 3x + 1
Rational Expressions
ALGEBRA 2 LESSON 9-4
Solutions
1. 2x2 – 3x + 1 = (2x – 1)(x – 1)
Check: (2x – 1)(x – 1) = 2x2 – 2x – 1x + 1
= 2x2 – 3x + 1
3. 5x2 + 6x + 1 = (5x + 1)(x + 1)
Check: (5x + 1)(x + 1) = 5x2 + 5x + 1x + 1
= 5x2 + 6x + 1
2. 4x2 – 9 = (2x)2 – 32 = (2x + 3)(2x – 3)
4. 10x2 – 10 = 10(x2 – 1) = 10(x2 – 12)
= 10(x + 1)(x – 1)
6. • = = =
8. • = =
÷ = • = = = or 1
÷ = • = = = 1 2 4 6
1 • 4
2 • 6
1 • 2 • 2
2 • 2 • 3 / 3 5 7
2 • 3
5 • 7 35 15 8
5 • 8
4 • 15
5 • 2 • 4
4 • 5 • 3
3 • 2
4 • 1
2 • 2 • 1
5. • = = =
7. • = = =
9. ÷ 4 = • = =
÷ = • = = = 3 8 5 6
3 • 5
8 • 6
8 • 3 • 2 16 / 2
8 • 2
3 • 16
1 • 16 1 4
5 • 1
8 • 4 32 9
9 • 4
16 • 3
3 • 3 • 4
4 • 4 • 3
9-4

4. Simplify . State any restrictions on the variable.
Rational Expressions
ALGEBRA 2 LESSON 9-4
Simplify State any restrictions on the variable.
x2 – 6x – 16
x2 + 5x + 6
= Factor the polynomials.
Notice x = –3 or –2. /
x2 – 6x – 16
x2 + 5x + 6
(x – 8)(x + 2)
(x + 3)(x + 2)
= Divide out common factors.
(x – 8)(x + 2)
(x + 3)(x + 2) 1 =
x – 8
x + 3
The simplified expression for x = –3 or –2.
x + 8
x + 3 /
The restrictions on x are needed to prevent the denominator of
the original expression from being zero.
Quick Check
9-4

5. Use the formulas for volume and surface area of a sphere.
Rational Expressions
ALGEBRA 2 LESSON 9-4
Quick Check
Compare the ratio of the volume to surface area of a sphere with radius r with the ratio of volume to surface area of a sphere with radius 2r.
Use the formulas for volume and surface area of
a sphere.
Volume (V) = • r 3
Surface Area (S.A.) = 4 r 2 4 3
Sphere with radius r Sphere with radius 2r
= = Write a ratio. V
S.A. 4 3
r 3
4 r 2 4 3
(r ) 3
4 (r )2
=      = Substitute for r.
(2r ) 3
4 (2r )2
= = Simplify. r 3 2r
The ratio of volume to surface area is twice as great for a sphere
whose radius is 2r than a sphere with a radius of r.
9-4

6. Multiply and . State any restrictions on the variable.
Rational Expressions
ALGEBRA 2 LESSON 9-4
Multiply and State any restrictions on the variable.
3x2 + 5x – 2
x – 5
x2 – 25
3x2 – 7x + 2
• = • Factor.
3x2 + 5x – 2
x – 5
x2 – 25
3x2 – 7x + 2
(3x – 1)(x + 2)
(x + 5)(x – 5)
(3x – 1)(x – 2)
= • Divide out
common
factors.
(3x – 1)(x + 2)
x – 5
(x + 5)(x – 5)
(3x – 1)(x – 2) 1 =
(x + 2)(x + 5)
x – 2 =
x2 + 7x + 10
x – 2
The product is for x 5, 2, or .
x2 + 7x + 10
x – 2 1 3 = /
Quick Check
9-4

7. Divide by . State any restrictions on the variables.
Rational Expressions
ALGEBRA 2 LESSON 9-4
Divide by . State any restrictions on the variables.
3 – y
(2x – 1)(x + 5)
6(y – 3)
(2x – 1)(x – 7)
÷ = • Multiply by the
reciprocal.
3 – y
(2x – 1)(x + 5)
6(y – 3)
(2x – 1)(x – 7)
= • Divide out
common factors.
3 – y
(2x – 1)(x + 5)
(2x – 1)(x – 7)
6(y – 3) 1 –1
= • –1
(x + 5)
(x – 7) 6 =
7 – x
6(x + 5)
= Multiply.
7 – x
6x + 30
The quotient is for x = –5, , or 7, and y = 3.
7 – x
6x + 30 / 1 2
Quick Check
9-4

8. Simplify. State any restrictions on the variable. 1. 2.
Rational Expressions
ALGEBRA 2 LESSON 9-4
Simplify. State any restrictions on the variable.
3. Multiply. State any restrictions on the variable. •
4. Divide. State any restrictions on the variable. ÷
x2 + x – 6
x2 + 3x
; x = –3, 0 /
x – 2 x
4x2– 25
2x2 + 3x – 20
; x = –4, /
2x + 5
x + 4 5 2
x2 + 6x – 7
x2 + 5x
3x2 + 16x + 5
2x2 + 7x – 9
; x = –5, – , 0, 1 /
3x2 + 22x + 7
2x2 + 9x 9 2
y 2 + 5y + 4
y 2 – 49
2y 2 + 5y – 12
y 2 + 9y + 14
; y = –7, –4, –2, , 7 / 3 2
y 2 + 3y + 2
2y 2 – 17y + 21
9-4