10-4
Multiplying and Dividing Rational Expressions
Preview
Warm Up
California Standards
Lesson Presentation
10-4
Multiplying and Dividing Rational Expressions
Warm Up
Multiply.
1. 2x2(x + 3) 2x3 + 6x2
3x2 – 8x – 35
2. (x – 5)(3x + 7)
3. 3x(x2 + 2x + 2)
4. Simplify
3x3 + 6x2 + 6x
Divide. Simplify your answer.
5.
6.
7.
8.
.
10-4
Multiplying and Dividing Rational Expressions
California
Standards
13.0 Students add, subtract, multiply, and
divide rational expressions and functions.
Students solve both computationally and
conceptually challenging problems by using
these techniques.
10-4
Multiplying and Dividing Rational Expressions
The rules for multiplying rational expressions
are the same as the rules for multiplying
fractions. You multiply the numerators, and you
multiply the denominators.
10-4
Multiplying and Dividing Rational Expressions
10-4
Multiplying and Dividing Rational Expressions
Additional Example 1A: Multiplying Rational
Expressions
Multiply. Simplify your answer.
Multiply the numerators and
denominators.
Factor.
Divide out the common factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 1B: Multiplying Rational
Expressions
Multiply. Simplify your answer.
Multiply the numerators and the
denominators. Arrange the
expression so like variables
are together.
Simplify.
Divide out common factors. Use
properties of exponents.
Simplify. Remember that z0 = 1.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 1C: Multiplying Rational
Expressions
Multiply. Simplify your answer.
Multiply. There are no common factors,
so the product cannot be simplified.
10-4
Multiplying and Dividing Rational Expressions
Remember!
Review the Quotient of Powers Property in
Lesson 7-4.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 1a
Multiply. Simplify your answer.
Multiply the numerators and the
denominators. Factor and
arrange the expression so like
variables are together.
Simplify.
Divide out common factors. Use
properties of exponents.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 1b
Multiply. Simplify your answer.
Multiply the numerators and the
denominators. Factor and
arrange the expression so like
variables are together.
Simplify.
Divide out common factors. Use
properties of exponents.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 2: Multiplying a Rational
Expression by a Polynomial
Multiply
answer.
. Simplify your
Write the polynomial over 1.
Factor the numerator and
denominator.
Divide out common factors.
Multiply remaining factors.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 2
Multiply
answer.
Simplify your
Write the polynomial over 1.
Factor the numerator and
denominator.
Divide out common factors.
Multiply remaining factors.
10-4
Multiplying and Dividing Rational Expressions
Remember!
Just as you can write an integer as a fraction,
you can write any expression as a rational
expression by writing it with a denominator of 1.
10-4
Multiplying and Dividing Rational Expressions
There are two methods for simplifying rational
expressions. You can simplify first by dividing out
and then multiply the remaining factors. You can
also multiply first and then simplify. Using either
method will result in the same answer.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 3: Multiplying a Rational
Expression Containing Polynomial
Multiply
answer.
. Simplify your
Method 1 Simplify first.
Factor.
Divide out common factors.
Then multiply.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 3 Continued
Method 2 Multiply first.
Multiply.
Distribute.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 3 Continued
Then simplify.
Factor.
Divide out common
factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 3a
Multiply
. Simplify your answer.
Simplify first.
Factor.
Divide out common factors.
Then multiply.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 3b
Multiply
. Simplify your answer.
Simplify first.
Factor.
Divide out common factors.
Then multiply.
p
Simplify.
10-4
Multiplying and Dividing Rational Expressions
The rules for dividing rational expressions are the
same as the rules for dividing fractions. To divide
by a rational expression, multiply by its reciprocal.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 4A: Dividing by Rational
Expressions and Polynomials
Divide. Simplify your answer.
Write as multiplication by the
reciprocal.
Multiply the numerators and the
denominators.
Divide out common factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 4B: Dividing by Rational
Expressions and Polynomials
Divide. Simplify your answer.
Write as multiplication by the
reciprocal.
Factor. Rewrite one opposite
binomial.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 4B Continued
Divide. Simplify your answer.
Divide out common factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 4C: Dividing by Rational
Expressions and Polynomials
Divide. Simplify your answer.
Write the binomial over 1.
Write as multiplication by the
reciprocal.
Multiply the numerators and the
denominators.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 4C Continued
Divide. Simplify your answer.
Divide out common factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 4a
Divide. Simplify your answer.
Write as multiplication by the
reciprocal.
Multiply the numerators and the
denominators.
Simplify. There are no common
factors.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 4b
Divide. Simplify your answer.
Write as multiplication by the
reciprocal.
Multiply the numerators and the
denominators and cancel
common factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 4c
Divide. Simplify your answer.
Write the trinomial over 1.
Write as multiplication by the
reciprocal.
Multiply.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 4c Continued
Divide. Simplify your answer.
Factor. Divide out common
factors.
Simplify.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 5: Application
Tanya is playing a carnival game. She needs to
pick 2 cards out of a deck without looking. The
deck has cards with numbers and cards with
letters. There are 6 more letter cards than
number cards.
a. Write and simplify an expression that represents the
probability that Tanya will pick 2 number cards.
Let x = the number cards.
number + letter = total
x
+ x + 6 = 2x + 6
Write expressions for
the number of each
kind of card and for
the total number of
items.
10-4
Multiplying and Dividing Rational Expressions
Additional Example 5 Continued
The probability of picking a number card and then
another number card is the product of the
probabilities of the individual events.
1st pick number
1st pick: total items
2nd pick number
2nd pick: total items
10-4
Multiplying and Dividing Rational Expressions
Additional Example 5 Continued
b. What is the probability that Tanya picks 2 number
cards if there are 25 number cards in the deck
before her first pick? Round your answer to the
nearest hundredth.
Since x represents the number of number cards,
substitute 25 for x.
P(number, number)
Substitute.
The probability is approximately 0.19.
Use the
order of
operations
to simplify.
10-4
Multiplying and Dividing Rational Expressions
Check It Out! Example 5
What if…? There are 50 blue items in the bag
before Marty’s first pick. What is the
probability that Marty picks two blue items?
Round your answer to the nearest hundredth.
Use the probability of picking two blue items. Since
x represents the number of blue items, substitute
50 for x.
P(blue, blue)
The probability is approximately 0.23.
Substitute.
Use the
order of
operations
to simplify.
10-4
Multiplying and Dividing Rational Expressions
Lesson Quiz: Part I
Multiply. Simplify your answer.
1.
2
2.
3.
Divide. Simplify your answer.
4.
5.
10-4
Multiplying and Dividing Rational Expressions
Lesson Quiz: Part II
6. A bag contains purple and green toy cars.
There are 9 more purple cars than green
cars.
a. Write and simplify an expression to represent the
probability that someone will pick a purple car
and a green car.
b. What is the probability of someone picking a
purple car and a green car if there are 12 green
cars before the first pick? Round to the nearest
hundredth.
0.24