7-4. } The sum of the coefficients of the outer and inner products is b. Going Deeper Essential question: How can you factor ax 2 + bx + c?

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1 Name Class Date 7-4 Factoring ax 2 + bx + c Going Deeper Essential question: How can you factor ax 2 + bx + c? You have learned how to factor a x 2 + bx + c when a = 1 by identifying the correct pair of factors of c whose sum is b. But what if the coefficient of x 2 is not 1? First, review binomial multiplication. The product (2x + 5)(3x + 2) is found by using FOIL. (2x + 5)(3x + 2) = 6 x 2 + 4x + 15x + 10 = 6 x x + 10 F O I L F The product of the coefficients of the first terms is a. O } The sum of the coefficients of the outer and inner products is b. I L The product of the last terms is c. To factor a x 2 + bx + c, you need to reverse this process. Start by listing the possible factor pairs of a and c. Then use trial and error to find a sum of b for the outer and inner products. 1 A-SSE.1.2 EXAMPLE Factoring ax 2 + bx + c Factor 5 n n + 2. A First list the possible factor pairs for both a and c. All of the signs of the terms are positive, so the factors of a and c must all be positive. The only factor pair for a is,. The only factor pair for c is,. B Choose the arrangement of the factor pairs that makes b = 11. Check your result by multiplying. REFLECT 5 n n + 2 = ( n + ) ( n + ) 1a. What other arrangement of factor pairs is possible for a and c? What is the resulting product, and how is it different from 5 n n + 2? 1b. If a is positive, b is negative, and c is positive, what are the signs of the factors of a and c that you are looking for? Chapter Lesson 4

2 1c. If a is positive, b is negative, and c is negative, what are the signs of the factors of a and c that you are looking for? If a and c have a lot of factors, there are many possible arrangements. One way to quickly check each arrangement is shown below, using the trinomial 5 n n + 2. List the factor pairs of a and c vertically, then multiply diagonally, and add. Factors Factors Inner and Outer of a of c products 1 2 = = 1 11 Sum If the sum is correct, the factors are read across: (1n + 2) and (5n + 1). 2 A-SSE.1.2 EXAMPLE Factoring ax 2 + bx + c Factor 6 x 2-13x - 8. A First list the possible factor pairs for both a and c. Because c is negative, one of the factors of c must be positive, and the other must be negative. The factor pairs for a are:, and,. The factor pairs for c are:, ;, ;, ;,. B Choose the arrangement of factor pairs that makes b = -13. Each factor pair of a can be arranged in two ways with each factor pair of c, so there are 16 possible arrangements. Three are shown below. 1 1 = 2 2 = 2 1 = 6-8 = 3-4 = 3-8 = REFLECT 6x 2-13x - 8 = ( x + ) ( x - ) 2a. If you know the factors of 6 x 2-13x - 8, how could you easily factor 6 x x - 8? 2b. What fact about the sign of the sum can you use so that you need to test at most half of the possible arrangements? Chapter Lesson 4

3 PRACTICE Factor. 1. 2x x z 2-30z x 2-10x d 2 + 7d g g y 2-2y n 2-11n a 2 + 7a x 2 - x 10. 9z h 2-12h n 2-20n x x y 2 + y - 18 To factor a polynomial of the form a x 2 + bx + c where a is negative, you first factor out -1 from all the terms. Factor each polynomial x x x 2-12x x 2 + 7x x 2 + 5x x 2 + 6x x 2-25x x x x 2 + 2x + 8 Chapter Lesson 4

4 23. A dolphin bounces a ball off its nose at an initial upward velocity of 6 m/s to a trainer lying on a 1-meter high platform. The polynomial -5 t 2 + 6t - 1 models the ball s height (in meters) above the platform. a. Factor the polynomial. b. When t = 0, what is the value of the polynomial? What does this value mean in the context of the situation? c. For what values on t does the polynomial equal 0? t = or t = d. Explain the two values for t in the context of the situation. Chapter Lesson 4

5 Name Class Date Additional Practice 7-4 Factor each trinomial. 1. 2x x x x x x x x x x x 2 23x x 2 59x x 2 14x x 2 73x x x x 2 + 2x x x x x x x x 2 3x x 2 7x x 2 49x x 2 + x x 2 35x x x x 2 + 5x The area of a rectangle is 20x 2 27x 8. The length is 4x + 1. What is the width? Chapter Lesson 4

6 Problem Solving 1. A rectangular painting has an area of ( ) cm 2. Its length is (2 2) cm. Find the width of the painting. 2. A ball is kicked straight up into the air. The height of the ball in feet is given by the expression , where is time in seconds. Factor the expression. Then find the height of the ball after 1 second. 3. Instructors led an exercise class from a raised rectangular platform at the front of the room. The width of the platform was (3 1) feet and the area was ( ) ft 2. Find the length of this platform. After the exercise studio is remodeled, the area of the platform will be ( ) ft 2. By how many feet will the width of the platform change? 5. The area of a soccer field is ( ) m 2. The width of the field is (4 10) m. What is the length? A (3 10) m C (6 10) m B (6 1) m D (8 2) m 7. For a certain college, the number of applications received after recruiting seminars is modeled by the polynomial What is this expression in factored form? A (3 40) ( 150) B (3 40) ( 150) C (3 30) ( 200) D (3 30) ( 200) 4. A clothing store has a rectangular clearance section with a length that is twice the width w. During a sale, the section is expanded to an area of ( ) ft 2. Find the amount of the increase in the length and width of the clearance section. 6. A square parking lot has an area of ( ) ft 2. What is the length of one side of the parking lot? F (2 5) ft H (5 4) ft G (2 10) ft J (5 2) ft 8. Jin needs to fence in his rectangular backyard. The fence will have one long section away from, but parallel to, the length of his house and two shorter sides connecting that section to the house. The length of Jin s house is (3 4) yd and the area of his backyard is ( ) yd 2. How many yards of fencing will Jin need? F (6 2) yd H (9 9) yd G (9 6) yd J (12 10) yd Chapter Lesson 4