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Section 8.3

Section 8.3. Factoring Trinomials: x ² + bx + c. Factor trinomials of the form x 2 + bx + c. Solve equations of the form x 2 + bx + c = 0. Factor x ² + bx + c. Observe the following pattern in this multiplication: ( x + 2)( x + 3) = x ² + (3 + 2) x + (2 ∙ 3)

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Section 8.3

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  1. Section 8.3 Factoring Trinomials: x² + bx + c

  2. Factor trinomials of the form x2 + bx + c. • Solve equations of the form x2 + bx + c = 0.

  3. Factor x² + bx + c Observe the following pattern in this multiplication: (x + 2)(x + 3) = x² + (3 + 2)x + (2 ∙ 3) (x + m)(x + n) = x² + (n+ m)x+ mn = x² + (n + m)x + mn x² + bx + c Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n.

  4. Factors of 12 Sum of Factors b and c are Positive Factor x2 + 7x + 12. In this trinomial, b = 7 and c = 12. You need to find two numbers with a sum of 7 and a product of 12. Make an organized list of the factors of 12, and look for the pair of factors with a sum of 7. 1, 12 13 2, 6 8 3, 4 7 The correct factors are 3 and 4. x2 + 7x + 12 = (x + m)(x + n) Write the pattern.

  5. b and c are Positive x2 + 7x + 12 = (x + m)(x + n) = (x + 3)(x + 4) m = 3 and n = 4 Answer: (x + 3)(x + 4) Check You can check the result by multiplying the two factors. F O I L(x + 3)(x + 4) = x2 + 4x + 3x + 12 FOIL method = x2 + 7x + 12 Simplify. 

  6. Factors of 27 Sum of Factors b is Negative and c is Positive Factor x2 – 12x + 27. In this trinomial, b = –12 and c = 27. This means m + n is negative and mn is positive. So m and n must both be negative. Make a list of the negative factors of 27, and look for the pair with a sum of –12. –1, –27 –28 –3, –9 –12 The correct factors are –3 and –9. x2 – 12x + 27 = (x + m)(x + n) Write the pattern.

  7. b is Negative and c is Positive = (x – 3)(x – 9) m= –3 and n = –9 x2 – 12x + 27 = (x + m)(x + n) Answer: (x – 3)(x – 9)

  8. c is Negative A. Factor x2 + 3x – 18. In this trinomial, b = 3 and c = –18. This means m + n is positive and mn is negative, so either m or n is negative, but not both. Therefore, make a list of the factors of –18 where one factor of each pair is negative. Look for the pair of factors with a sum of 3.

  9. Factors of –18 Sum of Factors c is Negative x2 + 3x – 18 1, –18 –17 –1, 18 17 2, –9 –7 –2, 9 7 3, –6 –3 –3, 6 3 The correct factors are –3 and 6. x2 + 3x – 18 = (x + m)(x + n) Write the pattern. Answer: = (x – 3)(x + 6) m = –3 and n = 6

  10. Factors of –20 Sum of Factors Solve an Equation by Factoring B. Factor x2 – x – 20. Since b = –1 and c = –20, m + n is negative and mn is negative. So either m or n is negative, but not both. 1, –20 –19 –1, 20 19 2, –10 –8 –2, 10 8 4, –5 –1 –4, 5 1 The correct factors are 4 and –5.

  11. Solve an Equation by Factoring x2 – x – 20 = (x + m)(x + n) Write the pattern. = (x + 4)(x – 5) m = 4 and n = –5 Answer: (x + 4)(x – 5)

  12. Solve an Equation by Factoring Solve x2 + 2x – 15 = 0. Check your solution. x2 + 2x – 15 = 0 Original equation (x + 5)(x – 3) = 0 Factor. x + 5 = 0 or x – 3 = 0 Zero Product Property x = –5 x = 3 Solve each equation. Answer: The solution set is {–5, 3}.

  13. ? ? ? ? (–5)2 + 2(–5) – 15 = 0 32 + 2(3) – 15 = 0 25 + (–10) – 15 = 0 9 + 6 – 15 = 0 Solve an Equation by Factoring Check Substitute –5 and 3 for x in the original equation. x2 + 2x – 15 = 0 x2 + 2x – 15 = 0 0 = 0 0 = 0  

  14. Homework Assignment #45 8.3 Skills Practice Sheet

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