Holt Algebra 1

8-3 Factoring x2 + bx + c

1.Factor quadratic trinomials of the form x2 + bx + c.

Objective

2. Factor quadratic trinomials of the form x2 + bx + c.

3. Factor four terms by grouping(front 2, back 2)

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 1A: Factoring by Using the GCFFactor each polynomial. Check your answer.

–14x – 12x2

Check –2x(7 + 6x)

Multiply to check your answer.

The product is the original polynomial.

–14x – 12x2

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 1B

Factor each polynomial. Check your answer.

8x4 + 4x3 – 2x2

8x4 = 2 2 2 x x x x4x3 = 2 2 x x x2x2 = 2 x x

2 x x = 2x2

4x2(2x2) + 2x(2x2) –1(2x2)

2x2(4x2 + 2x – 1)

Check 2x2(4x2 + 2x – 1)

8x4 + 4x3 – 2x2

The GCF of 8x4, 4x3 and –2x2 is 2x2.

Multiply to check your answer.

The product is the original polynomial.

Write terms as products using the GCF as a factor.

Use the Distributive Property to factor out the GCF.

Find the GCF.

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 2

Factor each expression.

a. 4s(s + 6) – 5(s + 6)

4s(s + 6) – 5(s + 6) The terms have a common binomial factor of (s + 6).

(4s – 5)(s + 6) Factor out (s + 6).

b. 7x(2x + 3) + (2x + 3)

7x(2x + 3) + (2x + 3)

7x(2x + 3) + 1(2x + 3)

(2x + 3)(7x + 1)

The terms have a common binomial factor of (2x + 3).

(2x + 1) = 1(2x + 1)

Factor out (2x + 3).

Holt Algebra 1

8-3 Factoring x2 + bx + c

You may be able to factor a polynomial by grouping. When a polynomial has four terms, you can make two groups and factor out the GCF from each group.

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 3A: Factoring by Grouping

Factor each polynomial by grouping. Check your answer.

6h4 – 4h3 + 12h – 8

(6h4 – 4h3) + (12h – 8)

2h3(3h – 2) + 4(3h – 2)

2h3(3h – 2) + 4(3h – 2)

(3h – 2)(2h3 + 4)

Group terms that have a common number or variable as a factor.

Factor out the GCF of each group.

(3h – 2) is another common factor.

Factor out (3h – 2).

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 3A Continued

Factor each polynomial by grouping. Check your answer.

Check (3h – 2)(2h3 + 4) Multiply to check your solution.

3h(2h3) + 3h(4) – 2(2h3) – 2(4)

6h4 + 12h – 4h3 – 8

The product is the original polynomial.

6h4 – 4h3 + 12h – 8

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 43B: Factoring by Grouping

Factor each polynomial by grouping. Check your answer.

5y4 – 15y3 + y2 – 3y

(5y4 – 15y3) + (y2 – 3y)

5y3(y – 3) + y(y – 3)

5y3(y – 3) + y(y – 3)

(y – 3)(5y3 + y)

Group terms.

Factor out the GCF of each group.

(y – 3) is a common factor.

Factor out (y – 3).

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 4: Factoring with Opposites

Factor 2x3 – 12x2 + 18 – 3x

2x3 – 12x2 + 18 – 3x

(2x3 – 12x2) + (18 – 3x)

2x2(x – 6) + 3(6 – x)

2x2(x – 6) + 3(–1)(x – 6)

2x2(x – 6) – 3(x – 6)

(x – 6)(2x2 – 3)

Group terms.

Factor out the GCF of each group.

Simplify. (x – 6) is a common factor.

Factor out (x – 6).

Write (6 – x) as –1(x – 6).

Holt Algebra 1

8-3 Factoring x2 + bx + c

Lesson Quiz: Part 1

Factor each polynomial. Check your answer.

1. 16x + 20x3

2. 4m4 – 12m2 + 8m

Factor each expression (by grouping).

3. 3y(2y + 3) – 5(2y + 3)

(2y + 3)(3y – 5)

4m(m3 – 3m + 2)

4x(4 + 5x2)

4. 2x3 + x2 – 6x – 3

5. 7p4 – 2p3 + 63p – 18

(2x + 1)(x2 – 3)

(7p – 2)(p3 + 9)

Holt Algebra 1

8-3 Factoring x2 + bx + c

(x + 2)(x + 5) = x2 + 7x + 10

Notice that when you multiply (x + 2)(x + 5), the constant term in the trinomial is the product of the constants in the binomials.

When you multiply two binomials, multiply:

First terms

Outer terms

Inner terms

Last terms

Remember!

Holt Algebra 1

8-3 Factoring x2 + bx + c

The guess and check method is usually not the most efficient method of factoring a trinomial. Look at the product of (x + 3) and (x + 4).

(x + 3)(x +4) = x2 + 7x + 12

x2 12

3x4x

The coefficient of the middle term is the sum of 3 and 4. The third term is the product of 3 and 4.

Holt Algebra 1

8-3 Factoring x2 + bx + c

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 1: Factoring x2 + bx + c

x2 + 6x + 5

Factor each trinomial. Check your answer.

(x + )(x + ) b = 6 and c = 5; look for factors of 5 whose sum is 6.

Factors of 5 Sum

1 and 5 6 The factors needed are 1 and 5.

(x + 1)(x + 5)

Check (x + 1)(x + 5) = x2 + x + 5x + 5 Use the FOIL method.The product is the

original polynomial.= x2 + 6x + 5

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 2: Factoring x2 + bx + c

Factor each trinomial. Check your answer.

x2 – 8x + 15

b = –8 and c = 15; look for factors of 15 whose sum is –8.

The factors needed are –3 and –5 .

Factors of –15 Sum–1 and –15 –16

–3 and –5 –8

(x – 3)(x – 5)

Check (x – 3)(x – 5 ) = x2 – 3x – 5x + 15 Use the FOIL method.The product is the

original polynomial.= x2 – 8x + 15

(x + )(x + )

Holt Algebra 1

8-3 Factoring x2 + bx + c

Example 3: Factoring x2 + bx + c

Factor each trinomial.

x2 + x – 20

(x + )(x + ) b = 1 and c = –20; look for factors of –20 whose sum is 1. The factor with the greater absolute value is positive.

The factors needed are +5 and –4.

Factors of –20 Sum

–1 and 20 19 –2 and 10 8

–4 and 5 1

(x – 4)(x + 5)

Holt Algebra 1

8-3 Factoring x2 + bx + c

Factor each trinomial. Check your answer.

Example 4

x2 + 2x – 15

(x + )(x + )

Factors of –15 Sum

–1 and 15 14 –3 and 5 2

(x – 3)(x + 5)

b = 2 and c = –15; look for factors of –15 whose sum is 2. The factor with the greater absolute value is positive.

The factors needed are –3 and 5.

Holt Algebra 1

8-3 Factoring x2 + bx + c

Lesson Quiz: Part I

Factor each trinomial.

1. x2 – 11x + 30

2. x2 + 10x + 9

3. x2 – 6x – 27

4. x2 + 14x – 32

(x + 16)(x – 2)

(x – 9)(x + 3)

(x + 1)(x + 9)

(x – 5)(x – 6)