Homework: Part I Multiply. Simplify your answer. 1. 2. Divide. Simplify your answer. 4.5. 3.

Homework: Part I

Multiply. Simplify your answer.

1. 2.

Divide. Simplify your answer.

4. 5.

3.

Homework: 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.

Warm UpMultiply.

1. 2x2(x + 3) 2. (x – 5)(3x + 7)

3. 3x(x2 + 2x + 2) 4. Simplify .

Divide. Simplify your answer.

5. 6. 7. 8.

2x3 + 6x23x2 – 8x – 35

3x3 + 6x2 + 6x

The rules for multiplying rational expressions are the same as the rules for multiplying fractions. You multiply the numerators, and you multiply the denominators.

Additional Example 1A: Multiplying Rational Expressions

Multiply. Simplify your answer.

Multiply the numerators and denominators.

Factor.

Divide out the common factors.

Simplify.

Additional Example 1B: Multiplying Rational ExpressionsMultiply. 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.

Additional Example 1C: Multiplying Rational Expressions

Multiply. Simplify your answer.

Multiply. There are no common factors, so the product cannot be simplified.

Review the Quotient of Powers Property in Lesson 7-4.

Remember!

Partner Share! 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.

Partner Share! 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.

Additional Example 2: Multiplying a Rational Expression by a Polynomial

Multiply . Simplify your answer.

Write the polynomial over 1.

Factor the numerator and denominator.

Divide out common factors.

Multiply remaining factors.

Partner Share! Example 2

Multiply Simplify your answer.

Write the polynomial over 1.

Factor the numerator and denominator.

Divide out common factors.

Multiply remaining factors.

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.

Additional Example 3: Multiplying a Rational Expression Containing Polynomial

Multiply . Simplify your answer.Method 1 Simplify first.

Then multiply.

Factor.

Divide out common factors.

Simplify.

Method 2 Multiply first.

Additional Example 3 Continued

Multiply.

Distribute.

Then simplify.

Factor.

Divide out common factors.

Simplify.

Additional Example 3 Continued

Multiply . Simplify your answer.

Simplify first.

Partner Share! Example 3a

Factor.

Divide out common factors.

Simplify.

Then multiply.

Multiply . Simplify your answer.

Simplify first.

Partner Share! Example 3b

Factor.

Divide out common factors.

Simplify.

Then multiply.

p

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.

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.

Additional Example 4B: Dividing by Rational Expressions and Polynomials

Divide. Simplify your answer.

Write as multiplication by the reciprocal.

Factor. Rewrite one opposite binomial.

Additional Example 4B Continued

Divide. Simplify your answer.

Divide out common factors.

Simplify.

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.

Additional Example 4C Continued

Divide. Simplify your answer.

Divide out common factors.

Simplify.

Divide. Simplify your answer.

Partner Share! Example 4a

Write as multiplication by the reciprocal.

Multiply the numerators and the denominators.

Simplify. There are no common factors.

Divide. Simplify your answer.Partner Share! Example 4b

Write as multiplication by the reciprocal.

Multiply the numerators and the denominators and cancel common factors.

Simplify.

Divide. Simplify your answer.

Partner Share! Example 4c

Write the trinomial over 1.

Write as multiplication by the reciprocal.

Multiply.

Divide. Simplify your answer.

Partner Share! Example 4c Continued

Factor. Divide out common factors.

Simplify.

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.

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 2nd pick number

1st pick: total items 2nd pick: total items

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.

Substitute.

Use the order of operations to simplify.

P(number, number)

The probability is approximately 0.19.

Partner Share! 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.

Substitute.

The probability is approximately 0.23.

P(blue, blue)

Use the order of operations to simplify.

Lesson Review: Part I

Multiply. Simplify your answer.

1. 2. 2

Divide. Simplify your answer.

4. 5.

3.

Lesson Review: 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