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Rational Expressions
Joseph Lee
Metropolitan Community College
Joseph Lee Rational Expressions
Domain of a Rational Expression
A rational expression will be defined as long as the denominatordoes not equal zero.
Joseph Lee Rational Expressions
Example 1.
State the domain of each rational expression.
x
x + 3
We are only concerned that the denominator x + 3 does not equal0.
x + 3 6= 0x 6= −3
The domain is all real numbers except x = −3.
Joseph Lee Rational Expressions
Example 1.
State the domain of each rational expression.
x
x + 3
We are only concerned that the denominator x + 3 does not equal0.
x + 3 6= 0x 6= −3
The domain is all real numbers except x = −3.
Joseph Lee Rational Expressions
Example 2.
State the domain of each rational expression.
2x + 3
3x − 2
We are only concerned that the denominator 3x − 2 does not equal0.
3x − 2 6= 03x 6= 2x 6= 2
3
The domain is all real numbers except x = 23 .
Joseph Lee Rational Expressions
Example 2.
State the domain of each rational expression.
2x + 3
3x − 2
We are only concerned that the denominator 3x − 2 does not equal0.
3x − 2 6= 03x 6= 2x 6= 2
3
The domain is all real numbers except x = 23 .
Joseph Lee Rational Expressions
Example 3.
State the domain of each rational expression.
x − 4
x2 + 5x + 6
We are only concerned that the denominator x2 + 5x + 6 does notequal 0.
x2 + 5x + 6 6= 0(x + 2)(x + 3) 6= 0
x 6= −2, x 6= −3
The domain is all real numbers except x = −2 and x = −3.
Joseph Lee Rational Expressions
Example 3.
State the domain of each rational expression.
x − 4
x2 + 5x + 6
We are only concerned that the denominator x2 + 5x + 6 does notequal 0.
x2 + 5x + 6 6= 0(x + 2)(x + 3) 6= 0
x 6= −2, x 6= −3
The domain is all real numbers except x = −2 and x = −3.
Joseph Lee Rational Expressions
Example 4.
State the domain of each rational expression.
x2 + 8x + 7
x2 + 1
We are only concerned that the denominator x2 + 1 does not equal0. In fact, for any value of x , x2 + 1 6= 0. Thus, the domain is allreal numbers.
Joseph Lee Rational Expressions
Example 4.
State the domain of each rational expression.
x2 + 8x + 7
x2 + 1
We are only concerned that the denominator x2 + 1 does not equal0. In fact, for any value of x , x2 + 1 6= 0. Thus, the domain is allreal numbers.
Joseph Lee Rational Expressions
Example 5.
Simplify. State any domain restrictions.
4x − 8
x − 2
4x − 8
x − 2=
4(x − 2)
x − 2
=x − 2
x − 2· 4
1
= 4, x 6= 2
Joseph Lee Rational Expressions
Example 5.
Simplify. State any domain restrictions.
4x − 8
x − 2
4x − 8
x − 2=
4(x − 2)
x − 2
=x − 2
x − 2· 4
1
= 4, x 6= 2
Joseph Lee Rational Expressions
Example 6.
Simplify. State any domain restrictions.
x − 3
x2 − 5x + 6
x − 3
x2 − 5x + 6=
x − 3
(x − 2)(x − 3)
=x − 3
x − 3· 1
x − 2
=1
x − 2, x 6= 2, 3
Joseph Lee Rational Expressions
Example 6.
Simplify. State any domain restrictions.
x − 3
x2 − 5x + 6
x − 3
x2 − 5x + 6=
x − 3
(x − 2)(x − 3)
=x − 3
x − 3· 1
x − 2
=1
x − 2, x 6= 2, 3
Joseph Lee Rational Expressions
Example 7.
Simplify. State any domain restrictions.
x2 − 14x + 49
x2 − 6x − 7
x2 − 14x + 49
x2 − 6x − 7=
(x − 7)2
(x − 7)(x + 1)
=x − 7
x − 7· x − 7
x + 1
=x − 7
x + 1, x 6= −1, 7
Joseph Lee Rational Expressions
Example 7.
Simplify. State any domain restrictions.
x2 − 14x + 49
x2 − 6x − 7
x2 − 14x + 49
x2 − 6x − 7=
(x − 7)2
(x − 7)(x + 1)
=x − 7
x − 7· x − 7
x + 1
=x − 7
x + 1, x 6= −1, 7
Joseph Lee Rational Expressions
Example 8.
Multiply. State any domain restrictions.
2x
3x + 1· 3x2 − 5x − 2
4x2 + 8x
2x
3x + 1· 3x2 − 5x − 2
4x2 + 8x=
2x
3x + 1· (3x + 1)(x − 2)
4x(x + 2)
=2x
2x· 3x + 1
3x + 1· x − 2
2(x + 2)
=x − 2
2(x + 2), x 6= −2,−1
3, 0
Joseph Lee Rational Expressions
Example 8.
Multiply. State any domain restrictions.
2x
3x + 1· 3x2 − 5x − 2
4x2 + 8x
2x
3x + 1· 3x2 − 5x − 2
4x2 + 8x=
2x
3x + 1· (3x + 1)(x − 2)
4x(x + 2)
=2x
2x· 3x + 1
3x + 1· x − 2
2(x + 2)
=x − 2
2(x + 2), x 6= −2,−1
3, 0
Joseph Lee Rational Expressions
Example 9.
Multiply. State any domain restrictions.
x2 + x − 6
4− x2· x
2 + 4x + 4
x2 + 4x + 3
x2 + x − 6
4− x2· x
2 + 4x + 4
x2 + 4x + 3=
(x + 3)(x − 2)
(2− x)(2 + x)· (x + 2)2
(x + 3)(x + 1)
=x + 3
x + 3· x − 2
2− x· x + 2
2 + x· x + 2
x + 1
= 1 · (−1) · 1 · x + 2
x + 1, x 6= −3,−2,−1, 2
= −x + 2
x + 1, x 6= −3,−2,−1, 2
Joseph Lee Rational Expressions
Example 9.
Multiply. State any domain restrictions.
x2 + x − 6
4− x2· x
2 + 4x + 4
x2 + 4x + 3
x2 + x − 6
4− x2· x
2 + 4x + 4
x2 + 4x + 3=
(x + 3)(x − 2)
(2− x)(2 + x)· (x + 2)2
(x + 3)(x + 1)
=x + 3
x + 3· x − 2
2− x· x + 2
2 + x· x + 2
x + 1
= 1 · (−1) · 1 · x + 2
x + 1, x 6= −3,−2,−1, 2
= −x + 2
x + 1, x 6= −3,−2,−1, 2
Joseph Lee Rational Expressions
Example 10.
Divide. State any domain restrictions.
2x2 − 9x − 5
2x2 − 13x + 15÷ 4x2 − 1
4x2 − 8x + 3
2x2 − 9x − 5
2x2 − 13x + 15÷ 4x2 − 1
4x2 − 8x + 3=
2x2 − 9x − 5
2x2 − 13x + 15· 4x2 − 8x + 3
4x2 − 1
=(2x + 1)(x − 5)
(2x − 3)(x − 5)· (2x − 3)(2x − 1)
(2x − 1)(2x + 1)
=2x + 1
2x + 1· x − 5
x − 5· 2x − 3
2x − 3· 2x − 1
2x − 1
= 1, x 6= −1
2,
1
2,
3
2, 5,
Joseph Lee Rational Expressions
Example 10.
Divide. State any domain restrictions.
2x2 − 9x − 5
2x2 − 13x + 15÷ 4x2 − 1
4x2 − 8x + 3
2x2 − 9x − 5
2x2 − 13x + 15÷ 4x2 − 1
4x2 − 8x + 3=
2x2 − 9x − 5
2x2 − 13x + 15· 4x2 − 8x + 3
4x2 − 1
=(2x + 1)(x − 5)
(2x − 3)(x − 5)· (2x − 3)(2x − 1)
(2x − 1)(2x + 1)
=2x + 1
2x + 1· x − 5
x − 5· 2x − 3
2x − 3· 2x − 1
2x − 1
= 1, x 6= −1
2,
1
2,
3
2, 5,
Joseph Lee Rational Expressions
Example 11.
Add. State any domain restrictions.x2 − 5x
x2 − 7x + 12+
3x − 3
x2 − 7x + 12
x2 − 5x
x2 − 7x + 12+
3x − 3
x2 − 7x + 12=
x2 − 5x + 3x − 3
x2 − 7x + 12
=x2 − 2x − 3
x2 − 7x + 12
=(x − 3)(x + 1)
(x − 3)(x − 4)
=x + 1
x − 4, x 6= 3, 4
Joseph Lee Rational Expressions
Example 11.
Add. State any domain restrictions.x2 − 5x
x2 − 7x + 12+
3x − 3
x2 − 7x + 12
x2 − 5x
x2 − 7x + 12+
3x − 3
x2 − 7x + 12=
x2 − 5x + 3x − 3
x2 − 7x + 12
=x2 − 2x − 3
x2 − 7x + 12
=(x − 3)(x + 1)
(x − 3)(x − 4)
=x + 1
x − 4, x 6= 3, 4
Joseph Lee Rational Expressions
Example 12.
Add. State any domain restrictions.x − 1
x2 + x − 20+
x + 3
x2 − 5x + 4
x − 1
x2 + x − 20+
x + 3
x2 − 5x + 4
=x − 1
(x + 5)(x − 4)+
x + 3
(x − 1)(x − 4)
=x − 1
(x + 5)(x − 4)· x − 1
x − 1+
x + 3
(x − 1)(x − 4)· x + 5
x + 5
=x2 − 2x + 1
(x + 5)(x − 4)(x − 1)+
x2 + 8x + 15
(x + 5)(x − 4)(x − 1)
=2x2 + 6x + 16
(x + 5)(x − 4)(x − 1), x 6= −5, 1, 4
Joseph Lee Rational Expressions
Example 12.
Add. State any domain restrictions.x − 1
x2 + x − 20+
x + 3
x2 − 5x + 4
x − 1
x2 + x − 20+
x + 3
x2 − 5x + 4
=x − 1
(x + 5)(x − 4)+
x + 3
(x − 1)(x − 4)
=x − 1
(x + 5)(x − 4)· x − 1
x − 1+
x + 3
(x − 1)(x − 4)· x + 5
x + 5
=x2 − 2x + 1
(x + 5)(x − 4)(x − 1)+
x2 + 8x + 15
(x + 5)(x − 4)(x − 1)
=2x2 + 6x + 16
(x + 5)(x − 4)(x − 1), x 6= −5, 1, 4
Joseph Lee Rational Expressions
Example 13.
Subtract. State any domain restrictions.4
x + 2− x − 26
x2 − 3x − 10
4
x + 2− x − 26
x2 − 3x − 10=
4
x + 2− x − 26
(x − 5)(x + 2)
=4
x + 2· x − 5
x − 5− x − 26
(x − 5)(x + 2)
=4x − 20
(x − 5)(x + 2− x − 26
(x − 5)(x + 2)
=(4x − 20)− (x − 26)
(x − 5)(x + 2)
=4x − 20− x + 26
(x − 5)(x + 2)
=3x + 6
(x − 5)(x + 2)
=3(x + 2)
(x − 5)(x + 2)
=3
x − 5, x 6= −2, 5
Joseph Lee Rational Expressions
Example 13.
Subtract. State any domain restrictions.4
x + 2− x − 26
x2 − 3x − 10
4
x + 2− x − 26
x2 − 3x − 10=
4
x + 2− x − 26
(x − 5)(x + 2)
=4
x + 2· x − 5
x − 5− x − 26
(x − 5)(x + 2)
=4x − 20
(x − 5)(x + 2− x − 26
(x − 5)(x + 2)
=(4x − 20)− (x − 26)
(x − 5)(x + 2)
=4x − 20− x + 26
(x − 5)(x + 2)
=3x + 6
(x − 5)(x + 2)
=3(x + 2)
(x − 5)(x + 2)
=3
x − 5, x 6= −2, 5
Joseph Lee Rational Expressions
Example 14.
Simplify the complex rational expression. State any domain
restrictions.1− 2
x
1− 4
x2
1− 2
x
1− 4
x2
=
1− 2
x
1− 4
x2
· x2x2
=x2 − 2x
x2 − 4
=x(x − 2)
(x − 2)(x + 2)
=x
x + 2, x 6= −2, 0, 2
Joseph Lee Rational Expressions
Example 14.
Simplify the complex rational expression. State any domain
restrictions.1− 2
x
1− 4
x2
1− 2
x
1− 4
x2
=
1− 2
x
1− 4
x2
· x2x2
=x2 − 2x
x2 − 4
=x(x − 2)
(x − 2)(x + 2)
=x
x + 2, x 6= −2, 0, 2
Joseph Lee Rational Expressions
Example 15.
Simplify the complex rational expression. State any domain
restrictions.
x
x + 3+ 2
x +2
x + 3
x
x + 3+ 2
x +2
x + 3
=
x
x + 3+ 2
x +2
x + 3
· x + 3
x + 3
=x + 2(x + 3)
x(x + 3) + 2
=x + 2x + 6
x2 + 3x + 2
=3x + 6
(x + 1)(x + 2)=
3
x + 1, x 6= −3,−2,−1
Joseph Lee Rational Expressions
Example 15.
Simplify the complex rational expression. State any domain
restrictions.
x
x + 3+ 2
x +2
x + 3x
x + 3+ 2
x +2
x + 3
=
x
x + 3+ 2
x +2
x + 3
· x + 3
x + 3
=x + 2(x + 3)
x(x + 3) + 2
=x + 2x + 6
x2 + 3x + 2
=3x + 6
(x + 1)(x + 2)=
3
x + 1, x 6= −3,−2,−1
Joseph Lee Rational Expressions
Example 16.
Simplify the complex rational expression. State any domain
restrictions.1 +
6
x+
9
x2
1− 1
x− 12
x2
1 +6
x+
9
x2
1− 1
x− 12
x2
=
1 +6
x+
9
x2
1− 1
x− 12
x2
· x2x2
=x2 + 6x + 9
x2 − x − 12
=(x + 3)2
(x − 4)(x + 3)
=x + 3
x − 4, x 6= −3, 0, 4
Joseph Lee Rational Expressions
Example 16.
Simplify the complex rational expression. State any domain
restrictions.1 +
6
x+
9
x2
1− 1
x− 12
x2
1 +6
x+
9
x2
1− 1
x− 12
x2
=
1 +6
x+
9
x2
1− 1
x− 12
x2
· x2x2
=x2 + 6x + 9
x2 − x − 12
=(x + 3)2
(x − 4)(x + 3)
=x + 3
x − 4, x 6= −3, 0, 4
Joseph Lee Rational Expressions