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**Warm Up Multiply. 1. 2x2(x + 3) 2. (x – 5)(3x + 7) 3x2 – 8x – 35**

3. 3x(x2 + 2x + 2) 4. Simplify Divide. Simplify your answer. 3x2 – 8x – 35 2x3 + 6x2 3x3 + 6x2 + 6x

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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.

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

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**Additional Example 1A: Multiplying Rational Expressions**

Multiply. Simplify your answer. Multiply the numerators and denominators. Factor. Divide out the common factors. Simplify.

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**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.

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**Additional Example 1C: Multiplying Rational Expressions**

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

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

Remember!

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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.

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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.

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**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.

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Check It Out! Example 2 Multiply Simplify your answer. Write the polynomial over 1. Factor the numerator and denominator. Divide out common factors. Multiply remaining factors.

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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. Remember!

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**There are two methods for simplifying 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.

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**Additional Example 3: Multiplying a Rational Expression Containing Polynomial**

Multiply Simplify your answer. Method 1 Simplify first. Factor. Divide out common factors. Then multiply. Simplify.

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**Additional Example 3 Continued**

Method 2 Multiply first. Multiply. Distribute.

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**Additional Example 3 Continued**

Then simplify. Factor. Divide out common factors. Simplify.

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**Divide out common factors.**

Check It Out! Example 3a Multiply Simplify your answer. Simplify first. Factor. Divide out common factors. Then multiply. Simplify.

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**Divide out common factors.**

Check It Out! Example 3b Multiply Simplify your answer. Simplify first. Factor. Divide out common factors. Then multiply. p Simplify.

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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.

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**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.

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**Additional Example 4B: Dividing by Rational Expressions and Polynomials**

Divide. Simplify your answer. Write as multiplication by the reciprocal. Factor. Rewrite one opposite binomial.

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**Additional Example 4B Continued**

Divide. Simplify your answer. Divide out common factors. Simplify.

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**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.

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**Additional Example 4C Continued**

Divide. Simplify your answer. Divide out common factors. Simplify.

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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.

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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.

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Check It Out! Example 4c Divide. Simplify your answer. Write the trinomial over 1. Write as multiplication by the reciprocal. Multiply.

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**Check It Out! Example 4c Continued**

Divide. Simplify your answer. Factor. Divide out common factors. Simplify.

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**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. Write expressions for the number of each kind of card and for the total number of items. number + letter = total x + x + 6 = 2x + 6

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**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

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**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. Use the order of operations to simplify. The probability is approximately 0.19.

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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) Substitute. Use the order of operations to simplify. The probability is approximately 0.23.

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Lesson Quiz: Part I Multiply. Simplify your answer. 1. 2 2. 3. Divide. Simplify your answer. 4. 5.

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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

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