Section 8.3Factoring Trinomials:

x² + bx + c

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

• Factor trinomials of the form x2 + bx + c.

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 nand the last term is the product of m and n.

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 are3 and 4.

Factors of 12 Sum of Factors

x2 + 7x + 12 = (x + m)(x + n) Write the pattern.

b and c are Positive

= (x + 3)(x + 4) m = 3 and n = 4

Check You can check the result by multiplying the two factors.

F O I L(x + 3)(x + 4) = x2 + 4x + 3x + 12FOIL method

= x2 + 7x + 12Simplify.

Answer: (x + 3)(x + 4)

x2 + 7x + 12 = (x + m)(x + n)

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.

Factors of 27 Sum of Factors

x2 – 12x + 27 = (x + m)(x + n) Write the pattern.

b is Negative and c is Positive

= (x – 3)(x – 9) m = –3 and n = –9

Answer: (x – 3)(x – 9)

x2 – 12x + 27 = (x + m)(x + n)

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.

c is Negative

x2 + 3x – 18 = (x + m)(x + n) Write the pattern.

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.

Answer: = (x – 3)(x + 6) m = –3 and n = 6

Factors of –18 Sum of Factors

x2 + 3x – 18

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.

Solve an Equation by Factoring

1, –20 –19

–1, 20 19

2, –10 –8

–2, 10 8

4, –5 –1

–4, 5 1 The correct factors are4 and –5.

Factors of –20 Sum of Factors

= (x + 4)(x – 5) m = 4 and n = –5

Solve an Equation by Factoring

x2 – x – 20 = (x + m)(x + n) Write the pattern.

Answer: (x + 4)(x – 5)

Solve x2 + 2x – 15 = 0. Check your solution.

Solve an Equation by Factoring

x2 + 2x – 15 = 0 Original equation

(x + 5)(x – 3) = 0 Factor.

Answer: The solution set is {–5, 3}.

x = –5 x = 3 Solve each equation.

x + 5 = 0 or x – 3 = 0 Zero Product Property

Solve an Equation by Factoring

Check Substitute –5 and 3 for x in the original equation.

x2 + 2x – 15 = 0 x2 + 2x – 15 = 0? ? (–5)2 + 2(–5) – 15 = 0 32 + 2(3) – 15 = 0

0 = 0 0 = 0

? ? 25 + (–10) – 15 = 0 9 + 6 – 15 = 0

Homework Assignment #45

8.3 Skills Practice Sheet