10-4 multiplying and dividing rational expressions warm up multiply. 1. 2x 2 (x + 3)2. (x – 5)(3x...
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10-4 Multiplying and Dividing Rational Expressions
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
10-4 Multiplying and Dividing Rational Expressions
13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
California Standards
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 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.
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
Review the Quotient of Powers Property in Lesson 7-4.
Remember!
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 . Simplify your answer.
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 Simplify your answer.
Write the polynomial over 1.
Factor the numerator and denominator.
Divide out common factors.
Multiply remaining factors.
10-4 Multiplying and Dividing Rational Expressions
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!
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 . Simplify your answer.Method 1 Simplify first.
Then multiply.
Factor.
Divide out common factors.
Simplify.
10-4 Multiplying and Dividing Rational Expressions
Method 2 Multiply first.
Additional Example 3 Continued
Multiply.
Distribute.
10-4 Multiplying and Dividing Rational Expressions
Then simplify.
Factor.
Divide out common factors.
Simplify.
Additional Example 3 Continued
10-4 Multiplying and Dividing Rational Expressions
Multiply . Simplify your answer.
Simplify first.
Check It Out! Example 3a
Factor.
Divide out common factors.
Simplify.
Then multiply.
10-4 Multiplying and Dividing Rational Expressions
Multiply . Simplify your answer.
Simplify first.
Check It Out! Example 3b
Factor.
Divide out common factors.
Simplify.
Then multiply.
p
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
Divide. Simplify your answer.
Check It Out! Example 4a
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
Divide. Simplify your answer.Check It Out! Example 4b
Write as multiplication by the reciprocal.
Multiply the numerators and the denominators and cancel common factors.
Simplify.
10-4 Multiplying and Dividing Rational Expressions
Divide. Simplify your answer.
Check It Out! Example 4c
Write the trinomial over 1.
Write as multiplication by the reciprocal.
Multiply.
10-4 Multiplying and Dividing Rational Expressions
Divide. Simplify your answer.
Check It Out! Example 4c Continued
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 2nd pick number
1st pick: total items 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.
Substitute.
Use the order of operations to simplify.
P(number, number)
The probability is approximately 0.19.
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.
Substitute.
The probability is approximately 0.23.
P(blue, blue)
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
Divide. Simplify your answer.
4. 5.
3.
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