10-3 Simplifying Rational Expressions
Preview
Warm Up
California Standards
Lesson Presentation
10-3 Simplifying Rational Expressions
Warm Up
Simplify each expression.
1.
2.
Factor each expression.
3. x2 + 5x + 6
4. 4x2 – 64
4(x + 4)(x – 4)
(x + 2)(x + 3)
5. 2x2 + 3x + 1
6. 9x2 + 60x + 100
(2x + 1)(x + 1)
(3x +10)2
10-3 Simplifying Rational Expressions
California
Standards
12.0 Students simplify fractions with
polynomials in the numerator and
denominator by factoring both and reducing
them to the lowest terms.
10-3 Simplifying Rational Expressions
Vocabulary
rational expression
10-3 Simplifying Rational Expressions
A rational expression is an algebraic
expression whose numerator and denominator
are polynomials. The value of the polynomial
expression in the denominator cannot be zero
since division by zero is undefined. This means
that rational expressions, like rational functions,
may have excluded values.
10-3 Simplifying Rational Expressions
Additional Example 1A: Identifying Excluded Values
Find any excluded values of each rational
expression.
g+4=0
g = –4
Set the denominator equal to 0.
Solve for g by subtracting 4 from
each side.
The excluded value is –4.
10-3 Simplifying Rational Expressions
Additional Example 1B: Identifying Excluded Values
Find any excluded values of each rational
expression.
x2 – 15x = 0
x(x – 15) = 0
Set the denominator equal to 0.
Factor.
x = 0 or x – 15 = 0 Use the Zero Product Property.
x = 0 or
x = 15 Solve for x.
The excluded values are 0 and 15.
10-3 Simplifying Rational Expressions
Additional Example 1C: Identifying Excluded Values
Find any excluded values of each rational
expression.
y2 + 5y + 4 = 0
(y + 4)(y + 1) = 0
Set the denominator equal to 0.
Factor.
y + 4 = 0 or y + 1 = 0 Use the Zero Product Property.
y = –4
or y = –1 Solve each equation for y.
The excluded values are –4 and –1.
10-3 Simplifying Rational Expressions
Remember!
To review the Zero Product Property,
see Lesson 9-5.
To review factoring trinomials, see Chapter 8.
10-3 Simplifying Rational Expressions
Check It Out! Example 1a
Find any excluded values of each rational
expression.
t2 + 5 = 0
Set the denominator equal to 0.
There are no values of t that make
the denominator equal to 0.
There are no excluded values.
10-3 Simplifying Rational Expressions
Check It Out! Example 1b
Find any excluded values of each rational
expression.
b2 + 5b = 0
b(b + 5) = 0
b = 0 or b + 5 = 0
b = –5
b = 0 or
Set the denominator equal to 0.
Factor.
Use the Zero Product Property.
Solve for b.
The excluded values are 0 and –5.
10-3 Simplifying Rational Expressions
Check It Out! Example 1c
Find any excluded values of each rational
expression.
k2 + 7k + 12 = 0
Set the denominator equal to 0.
(k + 4)(k + 3) = 0
Factor.
k + 4 = 0 or k + 3 = 0
Use the Zero Product Property.
k = –4
or k = –3 Solve each equation for k.
The excluded values are –4 and –3.
10-3 Simplifying Rational Expressions
A rational expression is in its simplest form
when the numerator and denominator have no
common factors except 1. Remember that to
simplify fractions, you can divide out common
factors that appear in both the numerator and
the denominator. You can do the same to
simplify rational expressions.
10-3 Simplifying Rational Expressions
Additional Example 2A: Simplifying Rational
Expressions
Simplify each rational expression, if possible.
Identify any excluded values.
4
Factor 14.
Divide out common factors.
Note that if r = 0, the
expression is undefined.
Simplify. The excluded value
is 0.
10-3 Simplifying Rational Expressions
Additional Example 2B: Simplifying Rational
Expressions
Simplify each rational expression, if possible.
Identify any excluded values.
3n; n ≠
Factor 6n² + 3n.
Divide out common factors.
Note that if n =
, the
expression is undefined.
Simplify. The excluded value
is
.
10-3 Simplifying Rational Expressions
Additional Example 2C: Simplifying Rational
Expressions
Simplify each rational expression, if possible.
Identify any excluded values.
3p – 2 = 0
3p = 2
There are no common factors.
Add 2 to both sides.
Divide both sides by 3. The
excluded value is
10-3 Simplifying Rational Expressions
Caution
Be sure to use the original denominator when
finding excluded values. The excluded values may
not be “seen” in the simplified denominator.
10-3 Simplifying Rational Expressions
Check It Out! Example 2a
Simplify each rational expression, if possible.
Identify any excluded values.
Factor 15.
Divide out common factors.
Note that if m = 0, the
expression is undefined.
Simplify. The excluded value
is 0.
10-3 Simplifying Rational Expressions
Check It Out! Example 2b
Simplify each rational expression, if possible.
Identify any excluded values.
Factor the numerator.
Divide out common factors. Note
that the expression is not
undefined.
Simplify. There is no excluded
value.
10-3 Simplifying Rational Expressions
Check It Out! Example 2c
Simplify each rational expression, if possible.
Identify any excluded values.
The numerator and denominator
have no common factors. The
excluded value is 2.
10-3 Simplifying Rational Expressions
From this point forward, you do not need to
include excluded values in your answers unless
they are asked for.
10-3 Simplifying Rational Expressions
Additional Example 3: Simplifying Rational
Expressions with Trinomials
Simplify each rational expression, if possible.
A.
Factor the numerator B.
and the denominator
when possible.
Divide out common
factors.
Simplify.
10-3 Simplifying Rational Expressions
Check It Out! Example 3
Simplify each rational expression, if possible.
a.
b.
Factor the numerator
and the denominator
when possible.
Divide out common
factors.
Simplify.
10-3 Simplifying Rational Expressions
Recall from Chapter 8 that opposite binomials can
help you factor polynomials. Recognizing opposite
binomials can also help you simplify rational
expressions.
Consider
The numerator and denominator
are opposite binomials. Therefore,
10-3 Simplifying Rational Expressions
Additional Example 4: Simplifying Rational
Expressions Using Opposite Binomials
Simplify each rational expression, if possible.
A.
B.
Factor.
Identify opposite
binomials.
Rewrite one
opposite binomial.
10-3 Simplifying Rational Expressions
Additional Example 4 Continued
Simplify each rational expression, if possible.
A.
B.
Divide out
common factors.
Simplify.
10-3 Simplifying Rational Expressions
Check It Out! Example 4
Simplify each rational expression, if possible.
a.
b.
Factor.
Identify opposite
binomials.
Rewrite one
opposite binomial.
10-3 Simplifying Rational Expressions
Check It Out! Example 4 Continued
Simplify each rational expression, if possible.
a.
b.
Divide out
common
factors.
Simplify.
10-3 Simplifying Rational Expressions
Check It Out! Example 4
Simplify each rational expression, if possible.
c.
Factor.
Divide out common factors.
Simplify.
10-3 Simplifying Rational Expressions
Additional Example 5: Application
A theater at an amusement park is shaped like
a sphere. The sphere is held up with support
rods.
a. What is the ratio of the theater’s volume to
its surface area? (Hint: For a sphere, V =
and S = 4r2.)
Write the ratio of volume to
surface area.
Divide out common factors.
10-3 Simplifying Rational Expressions
Additional Example 5 Continued
Use properties of exponents.
To divide by 4 multiply by the
reciprocal of 4.
Divide out common factors.
Simplify.
10-3 Simplifying Rational Expressions
Additional Example 5 Continued
b. Use this ratio to find the ratio of the
theater’s volume to its surface area when
the radius is 45 feet.
Write the ratio of volume to surface
area. Substitute 45 for r.
The ratio of volume to surface area of the
theater is 15:1.
10-3 Simplifying Rational Expressions
Check It Out! Example 5
Which barrel cactus has less of a chance to
survive in the desert, one with a radius of 6
inches or one with a radius of 3 inches?
Explain.
Write the ratio of surface to
volume twice. Substitute 6
and 3 for r.
Compare the ratios.
The barrel cactus with a radius of 3 inches has less
of a chance to survive. Its surface-area-to-volume
ratio is greater than for a cactus with a radius of 6
inches.
10-3 Simplifying Rational Expressions
Remember!
For two fractions with the same numerator, the
value of the fraction with a greater denominator
is less than the value of the other fraction.
9>3
10-3 Simplifying Rational Expressions
Lesson Quiz: Part I
Find any excluded values of each rational
expression.
1.
0
2.
0, 2
Simplify each rational expression, if possible.
3.
5.
4.
10-3 Simplifying Rational Expressions
Lesson Quiz: Part II
6. Calvino is building a rectangular tree house. The
length is 10 feet longer than the width. His friend
Fabio is also building a tree house, but his is
square. The sides of Fabio’s tree house are equal
to the width of Calvino’s tree house.
a. What is the ratio of the area of Calvino’s tree
house to the area of Fabio’s tree house?
b. Use this ratio to find the ratio of the areas if
the width of Calvino’s tree house is 14 feet.