Download Multiplying and Dividing Rational Expressions

Multiplying and Dividing
Rational Expressions
8-2
A rational expression is a quotient of two
polynomials.
5
x
A rational expression is undefined where the
denominator is equal to zero.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Example 1A: Simplifying Rational Expressions
Simplify. Identify any x-values for which the
expression is undefined.
10x8
6x4
510x8 – 4
5 x4
Quotient of Powers Property
=
3
36
The expression is undefined at x = 0 because
this value of x makes 6x4 equal 0.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Example 1B: Simplifying Rational Expressions
Simplify. Identify any x-values for which the
expression is undefined.
x2 + x – 2
x2 + 2x – 3
(x + 2)(x – 1) = (x + 2)
(x – 1)(x + 3)
(x + 3)
Factor; then divide out
common factors.
The expression is undefined at x = 1 and x = –3
because these values of x make the factors (x – 1)
and (x + 3) equal 0.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Example 1B Continued
Check Substitute x = 1 and x = –3 into the
original expression.
(1)2 + (1) – 2
0
=
(1)2 + 2(1) – 3
0
(–3)2 + (–3) – 2
4
=
(–3)2 + 2(–3) – 3
0
Both values of x result in division by 0, which is
undefined.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Check It Out! Example 1b
Simplify. Identify any x-values for which the
expression is undefined.
3x + 4
3x2 + x – 4
(3x + 4)
=
(3x + 4)(x – 1)
1
(x – 1)
Factor; then divide out
common factors.
The expression is undefined at x = 1 and x = – 4
3
because these values of x make the factors (x – 1)
and (3x + 4) equal 0.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Check It Out! Example 1b Continued
Check Substitute x = 1 and x = – 4
3 into
the original expression.
3(1) + 4
7
=
3(1)2 + (1) – 4
0
Both values of x result in division by 0, which is
undefined.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Example 2: Simplifying by Factoring by –1
2
4x
–
x
Simplify
. Identify any x values
2
x – 2x – 8
for which the expression is undefined.
–1(x2 – 4x)
x2 – 2x – 8
Factor out –1 in the numerator so that
x2 is positive, and reorder the terms.
–1(x)(x – 4)
(x – 4)(x + 2)
Factor the numerator and denominator.
Divide out common factors.
–x
(x + 2 )
Simplify.
The expression is undefined at x = –2 and x = 4.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Check It Out! Example 2a
10 – 2x . Identify any x values
x–5
for which the expression is undefined.
Simplify
–1(2x – 10)
x–5
Factor out –1 in the numerator so that
x is positive, and reorder the terms.
–1(2)(x – 5)
(x – 5)
Factor the numerator and denominator.
Divide out common factors.
–2
1
Simplify.
The expression is undefined at x = 5.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
You can multiply rational expressions the
same way that you multiply fractions.
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Example 3: Multiplying Rational Expressions
Multiply. Assume that all expressions are
defined.
5y3
3y4
3x
10x
A.

3
7
2x y
9x2y5
3
5
3x y3
2x3y7
5x3
3y5
Holt Algebra 2
5
3y4
10x

2 5
3 9x y
B.
x–3  x+5
4x + 20
x2 – 9
x–3 
x+5
4(x + 5) (x – 3)(x + 3)
1
4(x + 3)
8-2
Multiplying and Dividing
Rational Expressions
Check It Out! Example 3
Multiply. Assume that all expressions are
defined.
C.
2
10x – 40  x + 3
x2 – 6x + 8
5x + 15
10(x – 4)

(x – 4)(x – 2)
2
(x – 2)
Holt Algebra 2
x+3
5(x + 3)
8-2
Multiplying and Dividing
Rational Expressions
To divide rational expressions, multiply by the
reciprocal of the dividing expression.
5x4
15
÷
8x2y2 8y5
5x4
8y5

2
2
8x y
15
5x4
8y5

2
2
8x y
15
x2y3
3
Holt Algebra 2
Rewrite as multiplication
by the reciprocal.
8-2
Multiplying and Dividing
Rational Expressions
Example 4B: Dividing Rational Expressions
Divide. Assume that all expressions are
defined.
4 + 2x3 – 8x2
x4 – 9x2
x
÷
2
x – 4x + 3
x2 – 16
x4 – 9x2

2
x – 4x + 3
x2 – 16
x4 + 2x3 – 8x2
Rewrite as
multiplication by
the reciprocal.
x2 (x2 – 9) 
x2 – 16
x2 – 4x + 3
x2(x2 + 2x – 8)
x2(x – 3)(x + 3)  (x + 4)(x – 4)
(x – 3)(x – 1)
x2(x – 2)(x + 4)
(x + 3)(x – 4)
(x – 1)(x – 2)
Holt Algebra 2
8-2
Multiplying and Dividing
Rational Expressions
Check It Out! Example 4b
Divide. Assume that all expressions are
defined.
2x2 – 7x – 4 ÷
4x2– 1
x2 – 9
8x2 – 28x +12
2x2 – 7x – 4 
x2 – 9
8x2 – 28x +12
4x2– 1
(2x + 1)(x – 4)  4(2x2 – 7x + 3)
(x + 3)(x – 3)
(2x + 1)(2x – 1)
(2x + 1)(x – 4)  4(2x – 1)(x – 3)
(x + 3)(x – 3)
(2x + 1)(2x – 1)
4(x – 4)
(x +3)
Holt Algebra 2
Multiplying and Dividing
Rational Expressions
8-2
Check It Out! Example 5a
Solve.
x2 + x – 12
= –7
x+4
(x – 3)(x + 4) = –7
(x + 4)
x – 3 = –7
Note that x ≠ –4.
x = –4
Because the left side of the original equation is
undefined when x = –4, there is no solution.
Holt Algebra 2
Multiplying and Dividing
Rational Expressions
8-2
Check It Out! Example 5b
Solve.
4x2 – 9
=5
2x + 3
(2x + 3)(2x – 3) = 5
(2x + 3)
2x – 3 = 5
x=4
Holt Algebra 2
Note that x ≠ – 3 .
2
8-2
Multiplying and Dividing
Rational Expressions
HW pg. 580
#’s 19 – 31
HW pg. 580
#’s 15-17, 32-34, 42, 43, 45,46
Holt Algebra 2