ax 2 2

+

bx

+ +

c c

Warm Up

Lesson Presentation

Lesson Quiz

8-4

Factoring

ax 2

+

bx

+

c Warmup – Needed for L8-4 Factor each trinomial. Check your answer. HINT: Factor out the GCF first, then finish factoring

1. 3x 2 - 15x + 18 2. 5x 3 - 40x

2

+ 60x 3. 4x 2 – 18x + 14 4. 15x

3

+ 24x

2

+ 9x 3(x

–

2)(x

–

3) 5x(x – 6)(x

–

2) 2(2x – 7)(x – 1) 3x(5x + 3)(x + 1)

Holt Algebra 1

8-4

Factoring

ax 2

+

bx Warmup

+

c Factor by grouping:

1. b 3 - 2b – 8 + 4b 2 (b 2 – 2)(b + 4) 2. 2d 3 – d 2 – 3 + 6d (b 2 + 3)(2b - 1)

Factor each trinomial.

3. x 2 – 11x + 30 (x – 5)(x – 6) 4. x 2 + 10x + 9 (x + 1)(x + 9) 5. x 2 – 6x – 27 (x – 9)(x + 3)

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

Warm Up

Find each product.

1. (x – 2)(2x + 7) 2x

2

2. (3y + 4)(2y + 9) 6y 2 3. (3n – 5)(n – 7) 3n 2 + 3x

–

14 + 35y + 36 – 26n + 35

Factor each trinomial.

4. x 2 +4x – 32 5. z 2 + 15z + 36 6. h 2 – 17h + 72 (x – 4)(x + 8) (z + 3)(z + 12) (h – 8)(h – 9)

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

Objective

Factor quadratic trinomials of the form

ax

2

+ bx + c.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

In the previous lesson you factored trinomials of the form x 2 + bx + c. Now you will factor trinomials of the form ax + bx + c, where a ≠ 0.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

When you multiply (3x + 2)(2x + 5), the coefficient of the x 2 -term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials.

(

3

x +

2

)(

2

x +

5

) =

6

x

2

+ 19x +

10 Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

To factor a trinomial like ax binomial factors, write two sets of parentheses ( x + )( x + ). 2 + bx + c into its Write two numbers that are factors of a next to the

x

’ s and two numbers that are factors of c in the other blanks. Multiply the binomials to see if you are correct.

(

3

x +

2

)(

2

x +

5

) =

6

x

2

+ 19x +

10 Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 1: Factoring ax 2 + bx + c by Guess and Check Factor 6x 2 + 11x + 4 by guess and check.

( + )( + )

Write two sets of parentheses.

( x + )( x + ) The coefficient of the x the trinomial is 4.

2 ( 2 x + 4 )( 3

x +

1 ) = 6x 2 ( 1 x + 4 )( 6

x +

1 ) = 6x 2 ( 1 x + 2 )( 6

x +

2 ) = 6x 2 ( 1 x + 1 )( 6

x +

4 ) = 6x 2 ( 3 x + 4 )( 2

x +

1 ) = 6x 2

Holt Algebra 1

The first term is 6x 2 , so at least one variable term has a coefficient other than 1.

term is 6. The constant term in + 14x + 4  + 25x + 4  + 14x + 4  + 10x + 4  + 11x + 4 

Try factors of 6 for the coefficients and factors of 4 for the constant terms.

8-4

Factoring

ax 2

+

bx

+

c Example 1 Continued Factor 6x 2 + 11x + 4 by guess and check.

( + )( + )

Write two sets of parentheses.

( x + )( x + )

The first term is 6x 2 , so at least one variable terms has a coefficient other than 1.

The factors of 6x 2 + 11x + 4 are (3x + 4) and (2x + 1).

6x 2 + 11x + 4 = (3x + 4)(2x + 1)

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

So, to factor a 2 + bx + c, check the factors of a and the factors of c in the binomials. The sum of the products of the outer and inner terms should be b.

Product = a Product = c

( X + )( x + ) =

ax

2

+

bx

+

c Sum of outer and inner products = b Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

Since you need to check all the factors of a and the factors of c, it may be helpful to make a table. Then check the products of the outer and inner terms to see if the sum is b. You can multiply the binomials to check your answer.

Product = a Product = c

( X + )( x + ) =

ax

2

+

bx

+

c Sum of outer and inner products = b Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 2A: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer.

2x 2 + 17x + 21

( x + )( x + )

a = 2 and c = 21, Outer + Inner = 17.

Factors of 2 Factors of 21 1 and 2 1 and 2 1 and 2 1 and 2 1 and 21 21 and 1 3 and 7 7 and 3 Outer 1(21) 1(1) 1(7) 1(3) + Inner + 2(1) + 2(21) + 2(3) + 2(7) = 23 = 43

 

= 13 = 17

  (x + 7)(2x + 3)

Use the Foil method.

Check (x + 7)(2x + 3) = 2x 2 = 2x 2 + 3x + 14x + 21 + 17x + 21 

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Remember!

When b is negative and c is positive, the factors of c are both negative.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 2B: Factoring ax 2 + bx + c When c is Positive Factor each trinomial. Check your answer.

3x 2 – 16x + 16

( x + )( x + )

a = 3 and c = 16, Outer + Inner = –16 .

Factors of 3 Factors of 16 1 1 1 and 3 and 3 and 3 –1 – 4 Outer and –16 1(–16) – 2 and – 8 1( – 8) + 3( –2) and – 4 1( – 4) + Inner + 3( –1) = –19 = –14 + 3( – 4) = –16

   (x – 4)(3x Check (x – – 4) 4)(3x –

Use the Foil method.

4) = 3x 2 – 4x – 12x + 16 = 3x 2 – 16x + 16 

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

When c is negative, one factor of c will be positive and the other factor will be negative. Only some of the factors are shown in the examples, but you may need to check all of the possibilities.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 3A: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer.

3n 2 + 11n – 4

( y + )( y+ )

a = 3 and c = – 4, Outer + Inner = 11 .

Factors of 3 Factors of 4 1 1 and 3 and 3 1 and 3 1 and 3 –1 and –2 and –4 4 and 2 1 and –1 4 Outer 1(4) + Inner + 3( –1) 1(2) + 3( –2) = 1



= – 4 1(1) 1( –1) + 3( –4) = –11 + 3(4)

 (n + 4)(3n – 1) Check (n + 4)(3n –

Use the Foil method.

1) = 3n 2 – = 3n 2 n + 12n

–

+ 11n

–

4  4

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 3B: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer.

2x 2 + 9x – 18

( x + )( x+ )

Factors of 2 Factors of – 18 1 and 2 1 and 2 1 and 2 18 and 9 6 and and –1 –2 –3

a = 2 and c =

Outer 1( – 1) + Inner + 2(18)

–18, Outer + Inner = 9 .

(x + 6)(2x – 3)

Use the Foil method.

Check (x + 6)(2x – 3) = 2x 2 = 2x 2 – 3x + 12x

–

+ 9x

–

18  18

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 3C: Factoring ax 2 + bx + c When c is Negative Factor each trinomial. Check your answer.

4x 2 – 1 and 4 1 and 4 1 and 4 15x – 4

( x + )( x+ )

Factors of 4 Factors of – 4 –1 and –2 and –4 and 4 2 1

a = 4 and c = –4, Outer + Inner = –15.

Outer 1(4) + Inner – 1(4) = 0



1(2) – 2(4) 1(1) = –6



– 4(4) = –15

 (x – 4)(4x + 1)

Use the Foil method.

Check (x – 4)(4x + 1) = 4x 2 = 4x 2 + x – – 15x 16x

–

4

–

 4

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c

When the leading coefficient is negative, factor out – 1 from each term before using other factoring methods.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Caution

When you factor out –1 in an early step, you must carry it through the rest of the steps.

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Example 4A: Factoring ax 2 Negative + bx + c When a is Factor –2x 2 – 5x – 3.

–1 (2x 2 + 5x + 3) – 1( x + )( x+ )

Factors of 2 Factors of 3 1 and 2 1 and 2 3 and 1 1 and 3

(x + 1)(2x + 3) – 1(x + 1)(2x + 3)

Factor out –1. a = 2 and c = 3; Outer + Inner = 5

Outer + Inner 1(1) + 3(2) 1(3) + 1(2) = 5

 

Holt Algebra 1

8-4

Factoring

ax 2

+

bx

+

c Lesson Quiz Factor each trinomial. Check your answer.

1. 5x 2 + 17x + 6 (5x + 2)(x + 3) 2. 2x 2 + 5x – 12 3. 6x 2 – 23x + 7 (2x– 3)(x + 4) (3x – 1)(2x – 7) 4. –4x 2 + 11x + 20 5. 6x 2 + 14x + 4 (–x + 4)(4x + 5) 2(x + 2)(3x + 1)

Holt Algebra 1