# ESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c?

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1 LESSON 15.3 Factoring ax 2 + bx + c A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c? Factoring a x 2 + bx + c where c > 0 When you factor a polynomial in the form ax 2 + bx + c, the result will be the product of two binomial factors, in the form ( x + )( x + ). The product of the two coefficients of x will be a, and the product of the two constant terms will be c. The sum of the products of the inner and outer terms will be bx. Math On the Spot EXAMPLE 1 A.SSE.2 A Factor 4 x x STEP 1 Factor out any common factors of 4, 26, and x x + 42 = 2(2 x x + 21) STEP 2 STEP 3 Make a table that lists the factor pairs for a and c. Find the value of b that results from each combination of factor pairs. a = 2 c = 21 1 and 2 1 and 21 (1)(21) + (2)(1) = 23 1 and 2 3 and 7 (1)(7) + (2)(3) = 13 1 and 2 7 and 3 (1)(3) + (2)(7) = 17 1 and 2 21 and 1 (1)(1) + (2)(21) = 43 Use the combination of factor pairs that yields the correct value of b to factor the polynomial. (1x + 3)(2x + 7) =(x + 3)(2x + 7) 4 x x + 42 = 2(x + 3)(2x + 7) 13 is the sum that you re looking for. B Factor 3 x 2-26x STEP 1 Factor out any common factors of 3, 26, and 35. 3, 26, and 35 share no common factors other than 1. Lesson

2 STEP 2 Make a table that lists the factor pairs for a and c. Find the value of b that results from each combination of factor pairs. My Notes a = 3 c = 35 1 and 3-1 and -35 (1)(-35) + (3)(-1) = 38 1 and 3-5 and -7 (1)(-7) + (3)(-5) = 22 1 and 3-7 and -5 (1)(-5) + (3)(-7) = 26 1 and 3-35 and -1 (1)(-1) + (3)(-35) = 106 STEP 3 Use the combination of factor pairs that yields the correct value of b to factor the polynomial. -26 is the sum that you re looking for. 3x 2-26x + 35 = (1x - 7)(3x - 5) = (x - 7)(3x - 5) REFLECT 1. Critical Thinking When factoring 3 x 2-26x + 35, why should both factors of c be negative? 2. What If? If none of the factor pairs for a and c result in the correct value for b, what do you know about the polynomial? YOUR TURN Factor each polynomial x 2-14x x x x 2-48x x x + 70 Personal Math Trainer Online Practice and Help x x x 2-165x Unit 4

3 Factoring a x 2 + bx + c where c < 0 When factoring ax 2 + bx + c, if the value of c is negative, you know that one of the factors of c must be negative and one must be positive. Apply what you already know about factoring trinomials to this new situation. EXAMPLE 2 A.SSE.2 Math On the Spot A Factor 6 x 2-21x STEP 1 Factor out any common factors of 6, 21, and x 2-21x - 45 = 3(2 x 2-7x - 15) STEP 2 Make a table that lists the factor pairs for a and c. Since c is negative, one factor will be positive and the other will be negative. a = 2 c = and 2 1 and -15 (1)(-15) + (2)(1) = 13 1 and 2 3 and -5 (1)(-5) + (2)(3) = 1 1 and 2 5 and -3 (1)(-3) + (2)(5) = 7 1 and 2 15 and -1 (1)(-1) + (2)(15) = 29 1 and 2-1 and 15 (1)(15) + (2)(-1) = 13 1 and 2-3 and 5 (1)(5) + (2)(-3) = 1 1 and 2-5 and 3 (1)(3) + (2)(-5) = 7 1 and 2-15 and 1 (1)(1) + (2)(-15) = 29-7 is the sum that you re looking for. STEP 3 Use the combination of factor pairs that yields the correct value of b to factor the polynomial. (1x - 5)(2x + 3) = (x - 5)(2x + 3) B Factor 4 x 2 + 4x x 2-21x - 45 = 3(x - 5)(2x + 3) STEP 1 Factor out any common factors for 4, 4, and 35. STEP 2 4, 4, and 35 share no common factors other than 1. Make a table that lists the factor pairs for a and c. Since c is negative, one factor will be positive and the other will be negative. Lesson

4 Math Talk Mathematical Practices How does the sign of c help you choose the correct factor pair for c? a = 4 c = and 4 1 and -35 (1)(-35) + (4)(1) = 31 1 and 4 5 and -7 (1)(-7) + (4)(5) = 13 1 and 4 7 and -5 (1)(-5) + (4)(7) = 23 1 and 4 35 and -1 (1)(-1) + (4)(35) = and 4-1 and 35 (1)(35) + (4)(-1) = 31 1 and 4-5 and 7 (1)(7) + (4)(-5) = 13 1 and 4-7 and 5 (1)(5) + (4)(-7) = 23 1 and 4-35 and 1 (1)(1) + (4)(-35) = and 2 1 and -35 (2)(-35) + (2)(1) = 68 2 and 2 5 and -7 (2)(-7) + (2)(5) = 4 2 and 2 7 and -5 (2)(-5) + (2)(7) = 4 2 and 2 35 and -1 (2)(-1) + (2)(35) = 68 STEP 3 Use the combination of factor pairs that yields the correct value of b to factor the polynomial. 4x 2 + 4x - 35 = (2x + 7)(2x - 5) 4 is the sum that you re looking for. REFLECT 9. What If? Suppose a is a negative number. What would be the first step in factoring ax 2 + bx + c? Explain. 10. Make a Conjecture Using the information in the tables in Example 2, make a conjecture about what happens to b when you swap the positions of the plus and minus signs in the binomial factors. YOUR TURN Personal Math Trainer Online Practice and Help Factor each polynomial x x x x Unit 4

5 Guided Practice 1. Factor 3 x x (Example1) Complete the table for all factors of a and c. a = 3 c = 12 1 and 3 1 and 12 (1)(12) + (3)(1) = 1 and 3 2 and (1)( ) + ( )(2) = 12 1 and 3 and (1)( ) + (3)( ) = 1 and 3 and (1)( ) + ( )(4) = 1 and 3 and (1)( ) + ( )( ) = 1 and 3 and 1 (1)(1) + ( )( ) = 37 The factored form of 3 x x + 12 is (x + )( x + ). 2. Factor 8 x 2-2x - 6. (Example 2) 8, -2, and -6 have a common factor of, so 8 x 2-2x - 6 = (4 x 2 - x - 3) Complete the table for all factors of a and c. a = 4 c = -3? 1 and 4 1 and -3 (1)(-3) + (4)(1) = 1 1 and 3 and (1)( ) + (4)( ) = 1 and and (1)( ) + ( )( ) = 1 and and (1)( ) + ( )( ) = 2 and and (2)( ) + ( )(1) = and 3 and (2)( ) + (2)(3) = 4 The factored form of 8 x 2-2x - 6 is (x - )(4x + ). ESSENTIAL QUESTION CHECK-IN 3. How can you factor expressions of the form ax 2 + bx + c? Lesson

6 Name Class Date 15.3 Independent Practice A.SSE.2, A.SSE.3 Factor each trinomial, if possible x x - 15 Personal Math Trainer Online Practice and Help 13. The area of a soccer field is (24x x + 100) m 2. The width of the field is (4x + 10) m. What is the length? 5. 6x 2-29x x x z z d 2 + 7d Find all the possible values of b such that 3x 2 + bx - 2 can be factored. 15. Write the polynomial modeled, and then factor it. 12x 2 4x -15x y 2-11y g g n 2 + 3n Representing Real-World Problems The attendance at a team s basketball game can be approximated with the polynomial 5x x + 285, where x is the number of wins the team had in the previous month. a. Factor the polynomial completely. 12. How is factoring a trinomial in the form ax 2 + bx + c similar to factoring a trinomial in the form x 2 + bx + c? How is it different? b. Estimate the attendance if the team won 4 games in the previous month. 17. Kyle stood on a bridge and threw a rock up and over the side. The height of the rock, in meters, can be approximated by -5t 2 + 5t + 24, where t is the time in seconds after Kyle threw it. Completely factor the expression. 546 Unit 4

7 18. A triangle has an area of 1_ 2 ( 4x x + 30) ft 2. If the base of the triangle is (x + 6) ft, find the height of the triangle. 19. Draw Conclusions If a polynomial in the form ax 2 + bx + c has a = b = c = 1, can the expression be factored? Explain. 20. Counterexamples Marc thinks the only time a polynomial in the form ax 2 + bx + c cannot be factored is when at least one of the values for a, b, or c is a prime number. Find a counterexample to Marc s statement. 21. Shruti has a rectangular picture frame with an area of 30x 2 + 5x - 75 cm 2. a. Find the width of the frame if the height is (3x + 5) cm. b. Find the width of the frame if the height is (2x - 3) cm. c. Find the width of the frame when the height is 5 cm. 22. Communicate Mathematical Ideas Has the expression (3x + 7)(6x + 3) been factored completely? Explain. 23. Explain the Error Luna performed the work shown below to factor the polynomial 24x x + 3. Explain her error, and find the correctly factored form. 24x x + 3 = 3( 8x 2 + 6x + 0) = 3( 8x 2 + 6x) = 3(2x)(4x + 3) Lesson

8 24. The length of Rebecca s rectangular garden was two times the width, w. Rebecca increased the length and width of the garden so that the area of the new garden is ( 2w 2 + 7w + 6) square yards. By how much did Rebecca increase the length and the width? 25. The height in feet above the ground of a football that has been thrown or kicked can be described by the expression - 16t 2 + vt + h where t is the time in seconds, v is the initial upward velocity in feet per second, and h is the initial height in feet. a. Write an expression for the height of a football at time t when the initial upward velocity is 20 feet per second and the initial height is 6 feet. b. Factor your expression from part a. c. Find the height of the football after 1 second. FOCUS ON HIGHER ORDER THINKING Work Area 26. Critical Thinking Is there a value of m that will make x 2 + mx + 80 factorable? If so, how many? Explain and give all the possible values. 27. Explain the Error Frank has factored the polynomial 12x 2 + 5x - 2 as (3x - 1)(4x + 2). Explain his error. Give the correct factorization. 28. Communicate Mathematical Ideas Can the polynomial 4x 2 + 0x - 25 be factored? Explain. 548 Unit 4

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