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12.4 Simplify Rational Expressions
Warm Up
Lesson Presentation
Lesson Quiz
12.4 Warm-Up
Factor the polynomial.
1. x2 + 8x + 15
ANSWER
(x + 3)(x + 5)
2. 2x2 + 15x – 8
ANSWER
(x + 8)(2x – 1)
3. You pay $20 to join an aerobics center and pay
$4 per class session. Write an equation that gives
the average cost C per session as a function of the
number a of aerobics sessions that you take.
20
C
=
+4
ANSWER
a
12.4 Example 1
Find the excluded values, if any, of the expression.
a.
x+8
10x
b.
5
2y + 14
c.
4v
v2–9
d.
7w + 2
8w 2 + w + 5
SOLUTION
a.
The expression x + 8 is undefined when 10x = 0, or
10x
x = 0.
ANSWER
The excluded value is 0.
12.4 Example 1
SOLUTION
b. The expression
5 is undefined when
2y + 14
2y + 14 = 0, or x = –7.
ANSWER
The excluded value is –7.
12.4 Example 1
SOLUTION
c.
4v
is undefined when v2 – 9 = 0,
v2 – 9
or (v + 3)(v – 3) = 0. The solutions of the equation are
The expression
–3 and 3.
ANSWER
The excluded values are –3 and 3.
12.4 Example 1
SOLUTION
d. The expression
7w + 2
is undefined when
2
8w + w + 5
8w2 + w + 5 = 0.
The discriminant is b2 – 4ac = 12 – 4(8)(5) < 0. So, the
quadratic equation has no real roots.
ANSWER
There are no excluded values.
12.4 Guided Practice
Find the excluded values, if any, of the expression.
x+2
3x – 5
1.
ANSWER
The excluded value is 53
.
2
5y2 + 2y +3
2.
ANSWER
There are no excluded values.
12.4 Guided Practice
Find the excluded values, if any, of the expression.
n–6
3.
2n2 – 5n – 12
ANSWER
The excluded value is – 3 and 4 .
2
2m
m2 – 4
4.
ANSWER
The excluded value is –2, and 2 .
12.4 Example 2
Simplify the rational expression, if possible. State
the excluded values.
3 – 12m3
y
6m
r
5x
b.
a.
c.
d.
5(x + 2)
2r
7–y
18m2
SOLUTION
a. r = r
2r
2r
1
= 2
Divide out common factor.
Simplify.
ANSWER
The excluded value is 0.
12.4 Example 2
SOLUTION
b.
5 x
5x
=
5(x + 2) 5 (x + 2)
=
x
(x + 2)
Divide out common factor.
Simplify.
ANSWER
The excluded value is – 2.
12.4 Example 2
SOLUTION
6m2 (m – 2)
6 3 m2
6m2 (m – 2)
=
6 3 m2
c. 6m3 – 12m3 =
18m2
=
ANSWER
m–2
3
The excluded value is 0.
Factor numerator and
denominator.
Divide out common factors.
Simplify.
12.4 Example 2
SOLUTION
y
is already in simplest form.
d. The expression
7–y
ANSWER
The excluded value is 7.
12.4 Guided Practice
Simplify the rational expression, if possible. State
the excluded values.
3
4
a
5.
22a6
ANSWER
2
11a3
The excluded value is 0.
2c
c+5
ANSWER
2c
c+5
The excluded value is – 5.
2s2 + 8s
7.
3s +12
ANSWER
2s
3
The excluded value is – 4.
6.
8x
8.
ANSWER
3
2
8x + 16x
1
The excluded values are
x2 + 2x 0 and – 2.
12.4 Example 3
2 – 3x – 10
x
. State the excluded values.
Simplify 2
x + 6x + 8
SOLUTION
x2 – 3x – 10
=
x2 + 6x + 8
(x – 5)(x + 2)
(x + 4)(x + 2)
(x – 5)(x + 2)
(x + 4)(x + 2)
= x–5
x+4
=
Factor numerator and
denominator.
Divide out common factor.
Simplify.
ANSWER
The excluded values are – 4 and – 2.
12.4 Example 3
CHECK
In the graphing calculator activity
on page 560, you saw how to use a
graph to check a sum or difference
of polynomials.
Check your simplification using a
graphing calculator.
2 – 3x – 10
x
Graph y1 = 2
and y2 = x – 5
x + 6x + 8
x+4
The graphs coincide. So, the expressions are
equivalent for all values of x other than the
excluded values (–4 and –2).
12.4 Example 4
2 – 7x + 12
x
. State the excluded values.
Simplify
2
16 – x
SOLUTION
(x – 3)(x – 4)
x2 – 7x + 12
= (x – 4)(4 + x)
16 – x2
(x – 3)(x – 4)
=
– (x – 4)(4 + x)
(x – 3)(x – 4)
=
–(x – 4)(4+ x)
(x – 3)
(x – 3)
=–
=
(x + 4)
–(4 + x)
Factor numerator and
denominator.
Rewrite 4 – x as –( x – 4).
Divide out common factor.
Simplify.
ANSWER
The excluded values are –4 and 4.
12.4 Guided Practice
Simplify the rational expression. State the excluded
values.
x2 + 3x + 2
x2 + 7x + 10
ANSWER
(x + 1)
(x + 5)
The excluded values
are – 2 and – 5.
10.
y2 – 64
y2 – 16y + 64
ANSWER
(y + 8)
(y – 8)
The excluded value
is 8
11.
5 + 4z – z2
z2 – 3z – 10
– (z + 1)
ANSWER
(z + 2)
9.
The excluded values
are 5 and – 2.
12.4 Example 5
CELL PHONE COSTS
The average cost C (in dollars per
minute) for cell phone service in
the United States during the
period 1991–2000 can be modeled
by
46 – 2.2x
C=
100 – 18x + 2.2x2
where x is the number of years
since 1991. Rewrite the model so
that it has only whole number
coefficients.Then simplify the
model.
12.4 Example 5
SOLUTION
C=
46 – 2.2x
100 – 18x + 2.2x2
460 – 22x
=
1000 – 180x + 22x2
=
=
2(230 – 11x)
2(500 – 90x + 11x2)
Write model.
Multiply numerator and
denominator by 10.
Factor numerator and
denominator.
2(230 – 11x)
2(500 – 90x + 11x2)
230 – 11x
=
500 – 90x + 11x2
Divide out common
factor.
Simplify.
12.4 Guided Practice
12. In Example 5, approximate the average cost per
minute in 2000.
ANSWER
The average cost per minute in 2000 is
$.23/min.
12.4 Lesson Quiz
Simplify the rational expression, if possible. Find
the excluded values.
1.
12x3
18x5
ANSWER
2.
2 2
3 x ;0
2x – 14
x2 – 9x + 14
ANSWER
2
; 2, 7
x– 2
12.4 Lesson Quiz
3. The average cost C (in dollars per hour ) for a
cleaning service during the period 1994 – 2003
can be modeled by C = 52 – 1.4x ,where x is the
10 – 14x + 2.1x2
numbers of years since 1994. Rewrite the model so
that it has only whole number coefficients. Then
simplify the model, if necessary.
ANSWER
C=
520 – 14x
100 – 140x + 2x