Transcript Section 12.4 “Simplify Rational Expressions”

Objective

SWBAT simplify rational expressions, add,
subtract, multiply, and divide rational
expressions and solve rational equations
Section 12.3 “Dividing Polynomials”
Divide 5x² -10x +4 by -5x.
5 x 2  10x  4
 5x
“Think”
Simplify
 10 x
4
5x2
+
+
 5x
 5x
 5x
2
 10 x
4
5x
+
+
 5x
 5x
 5x
2
4
 x2
 5x
Solve
(6x³ + 3x² - 12x) ÷ 3x.
6 x  3x  12x
3x
3
2
3
“Think”
Simplify
6x
3x
2x²3
6x
3x
3x 2
+
3x
x
2
3x
+
3x
 12 x
+
3x
 12 x
+
3x
2x  x  4
2
-4
Section 12.4
“Simplify Rational Expressions”

A RATIONAL EXPRESSION is an
expression that can be written as a ratio (fraction)
of two polynomials where the denominator is not
0.
2
x3
7w  2
2
8w  w  5
Simplify Rational Expressions

To simplify a rational expression, you can factor the
numerator and denominator and then divide out
any common factors.
4 3 x
3
12 x


2
3
4 x x x
x
4x

A rational expression is in SIMPLEST FORM if
the numerator and denominator have no factors in
common other than 1.
Simplify the rational expression, if possible.
x  3x  10
x2  6x  8
2
Factor
( x  5)(x  2)
( x  4)(x  2)
x5
x4
Factor
5  4z  z 2
2
z  3z  10
(5  z )(z  1)
(5  z )(z  1)
( z  5)(z  2)  ( z  5)(z  2)
Recognize
opposites
Multiply by -1
Rewrite (z-5) as -(z+5)
z 1

z2
Section 12.5
“Multiply and Divide Rational Expressions”

Multiplying and dividing rational expressions is
similar to multiplying and dividing fractions.
a c ac
 
b d bd
a c a d ad
   
b d b c bc
Multiply by reciprocal,
then look to cancel terms.

Be sure to simplify your answer. Look to cancel
like terms when multiplying or dividing.
EXAMPLE 1
Find the product or quotient.
3
3
2 y 15y
 5
5y 8y
6
30y

6
40y
2
16r 12

3
5r
2
16r 5r

3 12
3
3

4
80r

36
20r 3

9
Multiply by reciprocal
EXAMPLE 2
Find the product.
3x  3x
x  4x  3

2
2
4 x  24x  36
x x
2
2
(3x  3x)(x  4 x  3)
2
2
(4 x  24x  36)(x  x)
2
2
3x( x  1)(x  3)(x  1)
4( x  3)(x  3) x( x  1)
3( x  1)
4( x  3)
Multiply numerator and denominator.
Factor and look for common
factors to cancel.
EXAMPLE 2
Find the quotient. Try it out.
7x2  7x
x 1
 2
2
x  2x  3 x  7x  8
7x2  7x x2  7x  8

2
x  2x  3
x 1
7 x( x  8)
( x  3)
Multiply by reciprocal
(7 x 2  7 x)(x 2  7 x  8)
( x 2  2 x  3)(x  1)
Multiply numerator
and denominator.
7 x( x  1)(x  8)(x  1)
( x  3)(x  1)(x  1)
Factor and look for common
factors to cancel.
EXAMPLE 3
Find the product.
5x
 ( x  3)
2
x  5x  6
5x
x3

2
x  5x  6 1
Multiply numerator and denominator.
5 x( x  3)
x 2  5x  6
Factor and look for common
factors to cancel.
5 x( x  3)
( x  3)(x  2)
5x
x2
Section 12.6
“Add and Subtract Rational Expressions”

Adding and subtracting rational expressions is
similar to adding and subtracting fractions.
a b ab
 
c c
c
a b a b
 
c c
c
Denominator must be COMMON!!!

Be sure to simplify your answer.
EXAMPLE 5
Find the sum or difference.
x4
x 1
 2
2
x  3x  10 x  2 x  8
x4
x 1

( x  5)(x  2) ( x  2)(x  4)
LCD = (x+5)(x – 2)(x + 4)
x4
( x  4)
x 1
( x  5)



( x  5)(x  2) ( x  4) ( x  2)(x  4) ( x  5)
( x  4)(x  4)  ( x  1)(x  5)

( x  5)(x  2)(x  4)
x  8x  16  ( x  4 x  5)
4 x  21


( x  5)(x  2)(x  4)
( x  5)(x  2)(x  4)
2
2
Section 12.7
“Solve Rational Equations”

A RATIONAL EQUATION is an equation that
contains one or more rational expressions.
6
x

x4 2

One method for solving a rational equation is to use the
cross products property. (You can use this method
when both sides of the equation are single rational
expressions).
Solve the equation. Check your solution.
5
y

y2 3
Cross multiply
(5)(3)  y( y  2)
15  y 2  2 y
0  y  2 y 15
2
0  ( y  5)( y  3)
y  5;3
Check for extraneous solutions
y=5
y = -3
5
y

y2 3
5
y

y2 3
5
5

52 3
5 5

3 3
5
3

3 2 3
5 3

5 3
Chapter 8
Final Review
12
6)
y6
64x12
2)
1
81a12 b 4
7)
3)
(7) 9
y10

27x 5
8)
4)
 36p 2
3 x12 y 5
4
5)
12d 3
c3
1)
x
9)
10)
d;
8
4 x 4
d;
1
144
Chapter 9
Final Review
 2a  9a  7
7)
 b3  2b 2  5b  6
2)
8x3  7 x 2  x  7
8)
2 x 3  11x 2  17x  20
3)
19n 2  4n  9
9)
4)
9c3  5c 2  5c  11
( x  7)(x  1)
5)
2 z 2  5z  63
1)
6)
2
144 72g  9 g 2
10)
11)
12)
(3x  4)(x  11)
5
a  0,
2
n  0,11,11
Chapter 10
Final Review
1; 3.5
1)
parabola
7)
2)
x  6,6
8)
A
3)
w  3,3
9)
D
10)
B
4)
 ( p  3)( p  1)
5)
1.1; 4.1
6)
1.11; 0.91
Chapter 11 Review worksheet
1)
4 2
8)
7 3 3 5
2)
4c 2 d 3 3c
9)
1 7 7
3)
4)
5)
6)
7)
no solution
6 10
10)
5
7
11)
x  11
12)
x7
13)
4 13
14)
d  53
15)
(3.5, 5)
6 3
8 7
2 15n
3n 2