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Factoring Trinomials

“Reverse FOIL”

Tobey & Slater, Intermediate Algebra, 6e 2

Trinomials of the Form x2 + bx + c

Using the FOIL method, we can show that

(x – 3)(x – 8)

x2 – 8x – 3x + 24

x2 – 11x + 24.

Therefore, by factoring (“reverse FOIL”) we obtain x2 – 11x + 24 = (x – 3)(x – 8).

Recall that factoring is the reverse process of multiplication.

Recognize

this form!

This trinomial results in the

product of two binomials.

Tobey & Slater, Intermediate Algebra, 6e 3

Trinomials of the Form x2 + bx + c

Factoring Trinomials of the Form x2 + bx + c

the answer will be of the form (x + m)(x + n)

Factoring Trinomials of the Form x2 + bx + c

the answer will be of the form (x + m)(x + n)

nx + mx = bx

nx

mx

x2 + bx + c

(x + m)(x + n)

FIRST terms come from the FIRST term.

LAST termsCome from The LAST term.

x . x = x2

Inside and outside terms combine to be the middle term.

m . n = c( )( ) (x )(x ( m)( n

Tobey & Slater, Intermediate Algebra, 6e 4

Trinomials of the Form x2 + bx + c

Example: Factor. x2 + 8x + 15

(x + 3)(x + 5)

Answer

x2 + 8x + 15

(x + 15)(x + 1)

1x

15x

1 15 8x x x

(x + 3)(x + 5)

5x

3x

5 3 8x x x

x . x15 x 1

or

5 x 3

or

Tobey & Slater, Intermediate Algebra, 6e 5

Facts About Signs

Facts About SignsSuppose x2 + bx + c = (x + m)(x + n). We know certain facts about m and n.

Facts About SignsSuppose x2 + bx + c = (x + m)(x + n). We know certain facts about m and n.

1. m and n have the same signs if c is positive. (Note: We did not say that they will have the same sign as c.)

a) They are positive if b is positive.

b) They are negative if b is negative.

2. m and n have opposite signs if c is negative. The larger number is positive if b is positive and negative if b is negative.

Tobey & Slater, Intermediate Algebra, 6e 6

Trinomials of the Form x2 + bx + c

Example: Factor. x2 + x – 42 One factor will be positive and one will be negative.

x2 + x – 42

( )( )

This form trinomialwill factor into two binomials.

Tobey & Slater, Intermediate Algebra, 6e 7

Try this.

Factor. x 2 – 9x + 8

Tobey & Slater, Intermediate Algebra, 6e 8

Trinomials of the Form ax2 + bx + c

Example: Factor. 3x2 + 4x + 1

(3x + 1)(x + 1)

3x2 + 4x + 13x.1x 1.1

3x

1x

1x + 3x = 4x

Answer

Tobey & Slater, Intermediate Algebra, 6e 9

Trial and Error method

Example: Factor. 9x2 13x + 4

1 . 4

2 . 2

(x – 1)(9x – 4)

1x . 9x

3x . 3x

The signs in both factors will be negative.

(x – 2)(9x – 2)(3x – 1)(3x – 4)

(3x – 2)(3x – 2)

– 13x

– 20x– 15x

– 12x

Correct

9x2 13x + 4

Tobey & Slater, Intermediate Algebra, 6e 10

Try this:

Factor.

a) 8x2 + 10x 3 b) 4x2 – 13x – 12