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Simplifying and Multiplying
Rational Expressions
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'
. Let a, b ' a ndCb e nonzer~ real nurn b ers or variabl~~:pres
~:;s"~ . 'f
;
ae = a ."c = !!
.i
.
_.-'
be
.
.
. b p' r
.
.
Divide out common factor c.
b
.
.
.
.
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.~...~,
Simplify a Rational Expression
..
<l
;t
. ~
.
...,.-.-
Simplify the rational expression, if possible.
a. 515x -_
3 .g"x
g'
= 3x
Divide out the common factor 5 and simplify.
x+5
b.--yc
. -14x2
6x
This expression is already in simplest form.
_ %.3.x
%. 7 'x, x
_ 3
__
7x
Divide out the common factors 2 and x.
J{~ll"~~,!(J/,Eipresslon
Simplify the expression, if Possible.
4x
8
1. 10
2. 4x
3
2!
12(x+ 1)
4. 1 -x
x
Simplify the expression, if possible.
5
.
~
2x
6 3x2(x+ 1)
. 3x
7. 2x2 + 8x
12x
8.~
3x + 6
Common Factors Write the common factors of the expressions.
16. 24x, 8
19. 3~,
16x4+ 56x2
17. 72x2, 18x
20. x2 - 81,x2 - lOx + 9
18. 4x2, 16x2 - 12x
21. x2 - 9, x2 - 4x - 12
Matching
Match the expression
22. 27x
9x2
with its simplified
23. 3x2 - 6x4
. 24.
~
form.
24x7 + 6x4
25.
15x7 + 9x3
5x5 + 3x
.,
A.3x2
Factoring and Simplifying Simplify the expression. If not possible,
write already in simplest form.
.
~
~
26. 15x
30. 2x4 + I
12x3
~
27. 4(x + I)
_4
~
29.
14x2 + 21x
42x4
31.
2
35. x
28. x2
+ 9x + 18
4x+24
36 x2+ 6x - 16
.
X
2
+ lOx -
24
x3 - x
37. x3 _ 7x2 + 6x
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Let a, b, and c be nonzero variable expressions.
To multiply, multiply numerators and denominators.
a . -.=.c
ac
--b
d
bd
Multiply Rational Expression~
.
Multiply 4x2
9~~.Simplify the result.
3x 8xSolution
4x2 9x
-8-=3x 8x5
36x3
24x6
Multiply the numerators
and denominators.
%.z.z-. 3 .~.~.~
2.Z.'z.Z-.X'.x.x. x. x. x
=~
Factor and divide out
the common factors.
Simplifythe expression.
2x3
MultlDlv Rational EXDressions
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I Solution
I 3x
Multiply
4x
~ 12 . ;
x2
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I
4x - 12
.
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I
I
I
I
(x - 3)(x + 3)
x
Multiply the numerators and
denominators.
= 3~~)(x
Divide out the common
factors.
4
CHECK
Factor the numerators and
denominators.
= 3x(x- 3)(x+ 3)
4x(x - 3)
= 3(x +
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'
3x
x- = 4(x- 3).
+ 3)
4~x~)
!
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x2 - 9
9. Simplify the result.
3)
Simplifythe expression.
Youcan check that your solutionis correct by substitutingthe samevalue
i for x, such as 1, in the original expression and in your answer.
J
3x
x2 - 9
3(1)
(1)2 - 9
3
-8
"
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.
4x - 12
x- = 4(1) - 12. (1)
3(x + 3) = 3(1 + 3) - 3 4 = 3 ,/
444
.
= -8 . T = 3
t
Multiply the rational expressions. Simplify the result.
9. 3x . E...
14 6x3
10. ~
. 2x - 4
3x - 6
x2
11. 4x + 8 . ~
x3
x2 - 4
Simplify
the Rational
result. ExpresSions
MUltiplying
39
5x. 1.
40
.2 x
42. -=L. 221
x +3 8(x- 1)
45.L
x+
1
. x2+x+2
3x + 2
MUltiply .the rational expressions.
7x2. 6x
. 3 4x2
43.~.~
5
.
41. 6x3 16
4x 2x2
8-6x
46. 5x + 45.
x2
3x3
x2
-
81
:
EXAMPLE
~ .
.]
.-'t'..,
!jj Multiply by a Polynomial
. 3(2 - x), Simplify the result.
Multiply x 2 -5xo+6
X
Solution
-
~
x2-5x+6
.3(2 1-
. 3(2_ x) = x2-5x+6
x
x)
Write polynomial as a
rational expression.
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=
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x
-1.3(x-2)
of
(2- x).
-3x(~)
~)(x
- 3)
Multiply. Then divide
out the common factor.
_ -3x
-x-3
I
Factor -lout
(x - 2)(x- 3) . ~
Simplify.
L-...
Multiply the expressions. Simplify the result.
12. x 13
Multiplying Polynomials
48. -x +x 5 (6x + 30)
.
52. 4x32x2
- 16x
.
(x2
. (2x + 6)
13. ,,_ 7
Multiply. Simplify the result.
16x9
+ 7x ~ 9)
49.
6x2
6
r5.
(24x - 8 )
53.
(15x
lOx
2x2 - 4x
,,,· (x2 + 4x)
-
70
-
5)
.(x2 - 49)
50. . (3 - 3x)
x2- I
5x+ 20 . (x2- 25)
51. x2-x+20
-
-0