Name Class Date 7-3 Factoring x 2 + bx + c Going Deeper Essential question: How can you factor x 2 + bx + c? 1 A-SSE.1.2 ENGAGE Factoring Trinomials You know how to multiply binomials: for example, (x + 3) (x - 5) = x 2-2x - 15. In this lesson, you will learn how to reverse this process and factor trinomials. There are several important things you should remember from multiplying binomials. Using FOIL, the constant term in the trinomial is a result of multiplying the last terms in the two binomials. Using FOIL, the x-term results from adding the products of the outside terms and inside terms. You can factor x 2 + 10x + 21 by working FOIL backward. Both signs in the trinomial are plus signs, so you know both binomials are of the form x plus something. Therefore, you can set up the factoring as shown below. x 2 + 10x + 21 = (x +? )(x +? ) To find the constant terms in the binomials, use the information above and follow the steps below. 1) The constant term in the trinomial, 21, is the product of the last terms in the two binomials. Factor 21 into pairs. The factor pairs are shown in the table at the right. 2) The correct factor pair is the one whose sum is the coefficient of x in the trinomial. 3) Complete the binomial expression with the appropriate numbers. x 2 + 10x + 21 = ( x + ) ( x + ) REFLECT Factors of 21 1a. You want to factor x 2-6x + 8. What factoring pattern would you set up to begin the process? Explain. Sum of Factors 1 and 21 22 3 and 7 10 Chapter 7 381 Lesson 3
1b. You want to factor x 2-2x - 15. What factoring pattern would you set up to begin the process? Explain. Would this pattern also work for x 2 + 2x - 15? Explain. 1c. Use factoring patterns to factor x 2 + 8x + 16 and x 2-6x + 9. What do you notice about the factored forms? What special type of trinomials are x 2 + 8x + 16 and x 2-6x + 9? 2 A-SSE.1.2 EXAMPLE Factoring Trinomials A Factor x 2 + 3x - 10. The constant is negative, so you know one binomial will have a subtraction sign. x 2 + 3x - 10 = (x +? )(x -? ) Complete the table at the right. Note that you are finding the factors of -10, not 10. Since the coefficient of x is positive, the factor with the greater absolute value will be positive (and the other factor will be negative). x 2 + 3x - 10 = ( x + ) ( x - ) Factors of -10-1 and 10 Sum of Factors B Factor x 2-8x - 48. The constant is negative, so you know one binomial will have a subtraction sign. x 2-8x - 48 = (x +? )(x -? ) Complete the table at the right. Since the coefficient of x is negative, the factor with the greater absolute value will be negative (and the other factor will be positive). x 2-8x - 48 = ( x + ) ( x - ) Factors of -48 1 and -48 2 and Sum of Factors Chapter 7 382 Lesson 3
REFLECT 2a. Complete the table below. Assume that b, c, p, and q are positive numbers. Trinomial Form of Binomial Factors x 2 + bx + c ( x p )( x q ) x 2 - bx + c ( x p )( x q ) x 2 - bx - c or x 2 + bx - c ( x p )( x q ) For the last row in the table, explain how to determine which factor contains a + sign and which factor contains a - sign. PRACTICE Complete the factorization of the polynomial. 1. t 2 + 6t + 5 = (t + 5) ( t + ) 2. z 2-121 = (z + 11) ( z ) 3. d 2 + 5d - 24 = ( d + ) ( d - ) 4. x 4-4 = ( x 2 + ) ( - 2 ) Factor the polynomial. 5. y 2 + 3y - 4 6. x 2-2x + 1 7. p 2-2p - 24 8. g 2-100 9. z 2-7z + 12 10. q 2 + 25q + 100 11. m 2 + 8m + 16 12. n 2-10n - 24 13. x 2 + 25x 14. y 2-13y - 30 Chapter 7 383 Lesson 3
Factor the polynomial. 15. z 2-9 16. p 2 + 3p - 54 17. x 2 + 11x - 42 18. g 2-14g - 51 19. n 2-81 20. y 2-25y 21. x 2 + 11x + 30 22. x 2 - x - 20 23. x 2 + 6x - 7 24. x 2 + 2x + 1 Chapter 7 384 Lesson 3
Name Class Date 7-3 Additional Practice Factor each trinomial. 1. x 2 + 7x + 10 2. x 2 + 9x + 8 3. x 2 + 13x + 36 4. x 2 + 9x + 14 5. x 2 + 7x + 12 6. x 2 + 9x + 18 7. x 2 9x + 18 8. x 2 5x + 4 9. x 2 9x + 20 10. x 2 12x + 20 11. x 2 11x + 18 12. x 2 12x + 32 13. x 2 + 7x 18 14. x 2 + 10x 24 15. x 2 + 2x 3 16. x 2 + 2x 15 17. x 2 + 5x 6 18. x 2 + 5x 24 19. x 2 5x 6 20. x 2 2x 35 21. x 2 7x 30 22. x 2 x 56 23. x 2 2x 8 24. x 2 x 20 25. Factor n 2 + 5n 24. Show that the original polynomial and the factored form describe the same sequence of numbers for n = 0, 1, 2, 3, and 4. n n 2 + 5n 24 n Chapter 7 385 Lesson 3
Problem Solving 1. A plot of land is rectangular and has an area of 2 5 24 m 2. The length is 3 m. Find the width of the plot. 3. The area of a poster board is 2 3 10 inches. The width is 2 inches. a. Write an expression for the length of the poster board. 2. An antique Persian carpet has an area of ( 2 20) ft 2 and a length of ( 5) feet. The rug is displayed on a wall in a museum. The wall has a width of ( 2) feet and an area of ( 2 17 30) ft 2. Write expressions for the length and width of both the rug and wall. Then find the dimensions of the rug and the wall if 20 feet. b. Find the dimensions of the poster board when 14. c. Write a polynomial for the area of the poster board if one inch is removed from each side. 4. The area of the addition is ( 2 10 200) ft 2. What is its length? A ( 20) feet B ( 2) feet C ( 2) feet D ( 20) feet 5. What is the area of the original house? F ( 2 10 200) ft 2 G ( 2 8 20) ft 2 H ( 2 12 200) ft 2 J ( 2 30 200) ft 2 6. The homeowners decide to extend the addition. The area with the addition is now ( 2 12 160) ft 2. By how many feet was the addition extended? A 1 foot C 3 feet B 2 feet D 4 feet Chapter 7 386 Lesson 3