8-4 Factoring ax 2 + bx + c Warm Up Warm Up Lesson Presentation Lesson Presentation California...

8-4 Factoring ax2 + bx + c

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California Standards

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Warm Up

Find each product.

1. (x – 2)(2x + 7)

2. (3y + 4)(2y + 9)

3. (3n – 5)(n – 7)

Find each trinomial.4. x2 + 4x – 325. z2 + 15z + 366. h2 – 17h + 72

6y2 + 35y + 36

2x2 + 3x – 14

3n2 – 26n + 35

(z + 3)(z + 12) (x – 4)(x + 8)

(h – 8)(h – 9)


11.0 Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

California Standards


In the previous lesson you factored trinomials of the form x2 + bx + c. Now you will factor trinomials of the form ax2 + bx + c, where a ≠ 0 or 1.


When you multiply (3x + 2)(2x + 5), the coefficient of the x2-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.

(3x + 2)(2x + 5) = 6x2 + 19x + 10


To factor a trinomial like ax2 + bx + c into its binomial factors, first write two sets of parentheses: ( x + )( x + ).

Write two integers that are factors of a next to the x’s and two integers that are factors of c in the other blanks. Then multiply to see if the product is the original trinomial. If there are no two such integers, we say the trinomial is not factorable.

8-4 Factoring ax2 + bx + cAdditional Example 1: Factoring ax2 + bx + c

Factor 6x2 + 11x + 4. Check your answer.

The first term is 6x2, so at least one variable term has a coefficient other than 1.

( x + )( x + )

The coefficient of the x2 term is 6. The constant term in the trinomial is 4.

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

(1x + 4)(6x + 1) = 6x2 + 25x + 4 (1x + 2)(6x + 2) = 6x2 + 14x + 4 (1x + 1)(6x + 4) = 6x2 + 10x + 4

(2x + 4)(3x + 1) = 6x2 + 14x + 4

(3x + 4)(2x + 1) = 6x2 + 11x + 4

8-4 Factoring ax2 + bx + cAdditional Example 1 Continued

Factor 6x2 + 11x + 4. Check your answer.

6x2 + 11x + 4 = (3x + 4)(2x + 1)

The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1).

Check (3x + 4)(2x + 1) = Use the FOIL

method.6x2 + 3x + 8x + 4

= 6x2 + 11x + 4The product of the original

trinomial.

8-4 Factoring ax2 + bx + cCheck It Out! Example 1a

Factor each trinomial. Check your answer.


( x + )( x + )

The coefficient of the x2 term is 6. The constant term in the trinomial is 3.

6x2 + 11x + 3

(1x + 3)(6x + 1) = 6x2 + 19x + 3 (1x + 1)(6x + 3) = 6x2 + 9x + 3


(2x + 1)(3x + 3) = 6x2 + 9x + 3

(3x + 1)(2x + 3) = 6x2 + 11x + 3

8-4 Factoring ax2 + bx + cCheck It Out! Example 1a Continued


The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3).

6x2 + 11x + 3 = (3x + 1)(2x +3)

Check (3x + 1)(2x + 3) = Use the FOIL method.

6x2 + 9x + 2x + 3

= 6x2 + 11x + 3

The product of the original trinomial.

8-4 Factoring ax2 + bx + cCheck It Out! Example 1b



( x + )( x + )

3x2 – 2x – 8

The coefficient of the x2 term is 3. The constant term in the trinomial is –8.


(1x – 1)(3x + 8) = 3x2 + 5x – 8

(1x – 8)(3x + 1) = 3x2 – 23x – 8 (1x – 4)(3x + 2) = 3x2 – 10x – 8

(1x – 2)(3x + 4) = 3x2 – 2x – 8


Factor each trinomial. Check your answer.3x2 – 2x – 8

The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4).

3x2 – 2x – 8 = (x – 2)(3x + 4)

Check (x – 2)(3x + 4) = Use the FOIL method.3x2 + 4x – 6x – 8

= 3x2 – 2x – 8 The product of the original trinomial.


So, to factor ax2 + 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.

( X + )( x + ) = ax2 + bx + c

Sum of outer and inner products = b

Product = cProduct = a


Since you need to check all the factors of a and all 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.

( X + )( x + ) = ax2 + bx + c

Sum of outer and inner products = b

Product = cProduct = a

8-4 Factoring ax2 + bx + cAdditional Example 2A: Factoring ax2 + bx + c When c

is PositiveFactor each trinomial. Check your answer.2x2 + 17x + 21

( x + )( x + )a = 2 and c = 21; Outer + Inner = 17.

(x + 7)(2x + 3)

Factors of 2 Factors of 21 Outer + Inner

1 and 2 1 and 21 1(21) + 2(1) = 23

1 and 2 21 and 1 1(1) + 2(21) = 43 1 and 2 3 and 7 1(7) + 2(3) = 13 1 and 2 7 and 3 1(3) + 2(7) = 17

Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21= 2x2 + 17x + 21

Use the FOILmethod.


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

Remember!



3x2 – 16x + 16a = 3 and c = 16, Outer + Inner = –16 .

(x – 4)(3x – 4)

Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16

= 3x2 – 16x + 16

Use the FOIL method.


1 and 3 –1 and –16 1(–16) + 3(–1) = –19 1 and 3 – 2 and – 8 1( – 8) + 3(–2) = –14 1 and 3 – 4 and – 4 1( – 4) + 3(– 4)= –16

( x + )( x + )

Additional Example 2B: Factoring ax2 + bx + c When c is Positive



6x2 + 17x + 5 a = 6 and c = 5; Outer + Inner = 17.


1 and 6 1 and 5 1(5) + 6(1) = 11

2 and 3 1 and 5 2(5) + 3(1) = 13 3 and 2 1 and 5 3(5) + 2(1) = 17

(3x + 1)(2x + 5)Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5

= 6x2 + 17x + 5


( x + )( x + )



9x2 – 15x + 4 a = 9 and c = 4; Outer + Inner = –15.


3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 3 and 3 – 2 and – 2 3(–2) + 3(–2) = –12 3 and 3 – 4 and – 1 3(–1) + 3(–4)= –15

(3x – 4)(3x – 1)

Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4

= 9x2 – 15x + 4


( x + )( x + )



3x2 + 13x + 12 a = 3 and c = 12; Outer + Inner = 13.


1 and 3 1 and 12 1(12) + 3(1) = 15 1 and 3 2 and 6 1(6) + 3(2) = 12 1 and 3 3 and 4 1(4) + 3(3) = 13

(x + 3)(3x + 4)

Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12

= 3x2 + 13x + 12


Check It Out! Example 2c

( x + )( x + )


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.

8-4 Factoring ax2 + bx + cAdditional Example 3A: Factoring ax2 + bx + c When c

is NegativeFactor each trinomial. Check your answer.

3n2 + 11n – 4

( n + )( n+ )

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

(n + 4)(3n – 1)Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4

= 3n2 + 11n – 4


Factors of 3 Factors of –4 Outer + Inner

1 and 3 –1 and 4 1(4) + 3(–1) = 1 1 and 3 –2 and 2 1(2) + 3(–2) = – 4 1 and 3 –4 and 1 1(1) + 3(–4) = –11

1 and 3 4 and –1 1(–1) + 3(4) = 11


Factor each trinomial. Check your answer.2x2 + 9x – 18

( x + )( x+ )a = 2 and c = –18;

Outer + Inner = 9 .


1 and 2 18 and –1 1(–1) + 2(18) = 35 1 and 2 9 and –2 1(–2) + 2(9) = 16 1 and 2 6 and –3 1(–3) + 2(6) = 9

(x + 6)(2x – 3)

Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18

= 2x2 + 9x – 18


Additional Example 3B: Factoring ax2 + bx + c When c is Negative



4x2 – 15x – 4

( x + )( x+ )a = 4 and c = –4;

Outer + Inner = –15.

Factors of 4 Factors of – 4 Outer + Inner

1 and 4 –1 and 4 1(4) – 1(4) = 0 1 and 4 –2 and 2 1(2) – 2(4) = –6 1 and 4 –4 and 1 1(1) – 4(4) = –15

(x – 4)(4x + 1) Use the FOIL method.

Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4

= 4x2 – 15x – 4

Additional Example 3C: Factoring ax2 + bx + c When c is Negative



6x2 + 7x – 3

( x + )( x+ )a = 6 and c = –3; Outer + Inner

= 7.


6 and 1 1 and –3 6(–3) + 1(1) = –17 6 and 1 3 and –1 6(–1) + 1(3) = – 3

3 and 2 3(–3) + 2(1) = –7 3 and 2 3(–1) + 2(3) = 3

1 and –3 3 and –1

2 and 3 2(–3) + 3(1) = –3 2 and 3 1(–2) + 3(3) = 7

1 and –3 3 and –1

(2x + 3)(3x – 1)Check (2x + 3)(3x – 1) = 6x2 – 2x + 9x – 3


= 6x + 7x – 3



4n2 – n – 3

( n + )( n+ )

a = 4 and c = –3; Outer + Inner = –1.

(n – 1) (4n + 3) Use the FOIL method.


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

Check (n – 1)(4n + 3) = 4n2 + 3n – 4n – 3

= 4n2 – n – 3


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


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

Caution!

8-4 Factoring ax2 + bx + cAdditional Example 4: Factoring ax2 + bx + c When

a is NegativeFactor –2x2 – 5x – 3.

–1(2x2 + 5x + 3)

–1( x + )( x+ )

Factor out –1. a = 2 and c = 3;

Outer + Inner = 5Factors of 2 Factors of 3 Outer + Inner

1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 2 1 and 3 1(3) + 1(2) = 5

–1(x + 1)(2x + 3)

(x + 1)(2x + 3)



–6x2 – 17x – 12 –1(6x2 + 17x + 12)

–1( x + )( x+ )

Factor out –1.

a = 6 and c = 12; Outer + Inner = 17


2 and 3 4 and 3 2(3) + 3(4) = 18 2 and 3 3 and 4 2(4) + 3(3) = 17

(2x + 3)(3x + 4) –1(2x + 3)(3x + 4) Check –1(2x + 3)(3x + 4) = –6x2 – 8x – 9x – 12

= –6x2 – 17x – 12



–3x2 – 17x – 10 –1(3x2 + 17x + 10)

–1( x + )( x+ )

Factor out –1. a = 3 and c = 10; Outer + Inner = 17)


1 and 3 2 and 5 1(5) + 3(2) = 11 1 and 3 5 and 2 1(2) + 3(5) = 17

(x + 5)(3x + 2)–1(x + 5)(3x + 2)Check –1(x + 5)(3x + 2) = –3x2 – 2x – 15x – 10

= –3x2 – 17x – 10

8-4 Factoring ax2 + bx + cLesson Quiz


1. 5x2 + 17x + 6

2. 2x2 + 5x – 12

3. 6x2 – 23x + 7

4. –4x2 + 11x + 20

5. –2x2 + 7x – 3

6. 8x2 + 27x + 9

(–x + 4)(4x + 5)

(3x – 1)(2x – 7)

(2x – 3)(x + 4)

(5x + 2)(x + 3)

(–2x + 1)(x – 3)

(8x + 3)(x + 3)

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