Download Simplifying Rational Expressions

Chapter 6 Section 1
SIMPLIFYING RATIONAL
EXPRESSIONS
Simplifying Rational Expressions
 A Rational Expression is an expression in the form
of qp where p and q are polynomials
and q ≠ 0. Exp:
x 1
3x  8
 A Rational Expressions are defined when the
denominator ≠ 0
 Set the denominator equal to 0 to determine if a
rational expression is defined.
Determine the values for which
rational expression are defined
Example:
x 1
3x  8
Find the values of x
that will make the
denominator 0.
Exclude these values.
3x  8  0
8
3   8
3
24
8
3
88
0
3x  8  8  0  8
3x  8
3x 8

3 3
8
x
3
Defined for all real numbers except
8
3
or
8
x
3
Determine the values for which
rational expression are defined
 Example:
x5
x 2  4 x  12
Factors of -12 add to -4
(-6)(2)=-12
x 2  4 x  12  0
( x  6)( x  2)  0
x  6  0 and x  2  0
x6
and x  2
(-6)+(2) = -4
Defined for all real numbers except 6 and -2
x  6 and x  2
Understanding the three signs of a fraction
a
a
means 
b
b
a
a
means 
b
b
a
a
 means 
b
b
Whenever a sign is omitted we assume it to be positive.
a
a a
 
b
b b
Changing any two of the three signs does
not change the value of a fraction
For a negative answer we will only use one negative sign in
the front of the fraction.
We generally write a fraction as 
a
b
or
-a
b
Do not put the negative in the denominator
Simplification of a Fraction
Example:
Example:
4
( 2 )(2)
2


6
3
( 2 )(3)
25
( 5 )(5)
5


40
8
( 5 )(8)
Simplification of a Fraction
Example:
78
( 2 )( 3 )(13)
13


192
32
( 2 )( 3 )(2)(2)(2)(2)(2)
Example:
18
(2)( 3 )( 3 )
2


45
5
(5)( 3 )( 3 )
Simplify Rational Expressions
 Factor both the numerator and the denominator as
completely as possible.
 Divide out any factors common to both the numerator
and the denominator. Simplify or reduce to lowest
term.
 You should ask yourself “Is the numerator and the
denominator factored completely?”
Simplify Rational Expressions
 Example:
9
3  3
3


12 3  2  2 4
 Example:
ab  b
2b
2
( b )(a  b)


(2)( b )
a b
2
Simplify Rational Expressions
 Example:
6 x  12 x  24 x
18 x3
4
3
Factors of -4 that adds to 2
(-1)(4), (1)(-4), (-2)(2)
PRIME
2

( 6 x )( x  2 x  4)
2
2
2
( 6 x )(3x)

2
x
  2x  4
 3 x 
Simplify Rational Expressions
 Example:
Factors of -10 that adds to 3
(-2)(5) = -10
(-2)+(5) = 3
x 2  3x  10
( x  5)( x  2 )


x2
( x2)
x5
 x5
1
Simplify Rational Expressions
 Example:
Difference in Two Squares: a2 - b2 = (a + b)(a – b)
a=m
and
b=7
m2 – 72 = (m + 7)(m – 7)
m2  49
(m  7)( m  7 )

 m7
m7
( m7)
Simplify Rational Expressions
 Example:
(a)(c) = (2)(-5) = -10 Factors of -10 that adds to 3
(-2)(5) = -10 and (-2)+(5) = 3 replace the 3x with -2x
and 5x
2x2 – 2x + 5x - 5 factor by grouping.
2 x 2  3x  5
(2 x  5)( x  1 )
2x  5


2
x  2x  3
x3
( x  3)( x  1 )
Factors of -3 that adds to 2
(-1)(3) = -3
(-1)+(3) = 2
Factor a -1 from a Polynomial
When -1 is factored out the sign of each term
changes.
Use when the numerator and denominator differ
only by their signs.
Example:
2x 1
2x 1
( 2 x 1 )
1



 1
1 2x
2 x  1
1
1( 2 x  1 )
Factor a -1 from a Polynomial
Example:
(a)(c) = (3)(4) = 12 Factors of 12 that adds to -8
(-6)(-2) = 12 and (-6)+(-2) = -8 replace the -8z with
-2z and -6z
3z2 – 6z – 2z + 4 factor by grouping.
3z 2  8 z  4
(3z  2)( z  2)
(3 z  2)( z  2)


2 z
(2  z )
( z  2)
(3z  2) or - 3 z  2

(3 z  2)( z  2 )
1( z  2 )
Remember
 Only factor can be cancelled, not terms.
 Factor completely before you try to simplify.
 When factoring a -1 the sign of each term changes
 Use -1 only when the numerator and denominator
differ by their signs.
HOMEWORK 6.1
Page 359:
#29, 31, 35, 39, 45, 47, 55, 57