Algebra I NotesSection 9.6 (A) Factoring ax2 + bx + c With Leading Coefficient

≠ 1

In section 9.5, we learned how to factor quadratic polynomials whose leading coefficient = 1. We will now learn how to factor quadratic polynomials whose leading coefficient is ≠ 1. To factor quadratic polynomials when a ≠ 1, we need to find the factors of a and the factors of c such that when we add the Outer and Inner products of the FOIL method, their sum = b. Example – Factor the polynomials We must find the factors of a and c 1. 3x2 – 17x + 10 - the factors of 3 (or a) are: ___________

- the factors of 10 (or c) are : ___________ We know that the binomials (without their signs) must be one of these four forms because of the factors: 1. 2. 3. 4.

1 , 31, 10 & 2, 5

(x 1)(3x 10) (x 10)(3x 1) (x 2)(3x 5) (x 5)(3x 2)

If we look at the FOIL method for each of the possible factors, we get:

F O I L 1. + + + 2. + + + 3. + + + 4. + + +

We know that in the original polynomial 3x2 – 17x + 10, 10 (or c) must be positive, so the L multiplication of the FOIL method must yield a + 10 (for c). At the same time, the sum of the O and I multiplication must yield a - 17 (for b).

With the correct choice of signs, which sum (of O and I) can possibly add up to -17? _______________

So we conclude: 3x2 – 17x + 10 = ____________________

3x2 10x 3x 10 3x2 x 30x 103x2 5x 6x 10

3x2 2x 15x 10

-2x & -15x

(x – 5)(3x – 2)

2. 3x2 – 4x – 7 - the factors of 3 (or a) are: _____________ - the factors of 7 (or c) are : _____________

We know that the binomials (without their signs) must be one of these two forms because of the factors: 1. 2.

If we look at the FOIL method for each of the possible factors, we get:

F O I L 1. + + + 2. + + + We know that in the original polynomial 3x2 – 4x – 7, -7 (or c) must be negative, so the L multiplication of the FOIL method must yield a -7 (for c). At the same time, the sum of the O and I multiplication must yield a - 4 (for b).

With the correct choice of signs, which sum (of O and I) can possibly add up to -4? ________________

So we conclude: 3x2 – 4x – 7 = ________________

1, 31, 7

(x 1)(3x 7) (x 7)(3x 1)

7x 3x

x 21x

-7x & 3x

(x + 1)(3x – 7)

3. 6x2 – 2x – 8 - begin by factoring out the common factor ___________- the factors of 3 (or a) are: _______________ - the factors of 4 (or c) are : _______________

We know that the binomials (without their signs) must be one of these three forms because of the factors: 1. 2. 3.

If we look at the FOIL method for each of the possible factors, we get:

F O I L 1. + + + 2. + + + 3. + + +

We know that in the FACTORED polynomial 3x2 – x – 4, -4 (or c) must be negative, so the L multiplication of the FOIL method must yield a -4 (for c). At the same time, the sum of the O and I multiplication must yield a - 1 (for b).

21, 32(3x2 – x – 4)1, 4 & 2, 2

(x 1)(3x 4) (x 4)(3x 1) (x 2)(3x 2)

4x 3xx 12x2x 6x

With the correct choice of signs, which sum (of O and I) can possibly add up to -1? ________________So we conclude: 2(3x2 – x – 4) = _______________________

-4x & 3x2(x + 1)(3x – 4)

More Examples - Factor the polynomials. 1. 3t2 + 16t + 5 2. 6b2 – 11b – 2

Factors of a : 1, 3

Factors of c : 1, 5

(t 1)(3t 5) (t 5)(3t 1)

5t , 3t t , 15t

Must add to 16t :

(t + 5)(3t + 1)

Factors of a : 1, 6 2, 3

Factors of c : 1, 2

(2b 1)(3b 2) (2b 2)(3b 1)

4b , 3b 2b , 6b

(b 1)(6b 2) (b 2)(6b 1)

2b , 6b b , 12b

Must add to -11b :

(b – 2)(6b + 1)

3. 5w2 – 9w – 2 4. 4x2 – 6x – 4

Factors of a : 1, 5

Factors of c : 1, 2

(w 1)(5w 2) (w 2)(5w 1)

2w , 5w w , 10w

Must add up to -9w :

(w - 2)(5w + 1)

Factors of a : 1, 2 Factors of c : 1, 2

Factor out GCF : 2

2(2x2 – 3x – 2)

(x 1)(2x 2) (x 2)(2x 1)

2x , 2x x , 4x

Must add up to -3x :

2(x – 2)(2x + 1)

5. 12y2 – 22y – 20 6. 9x2 + 21x – 18

Factor out GCF : 2

2(6y2 – 11y – 10)

Factors of a : 1, 6 2, 3

Factors of c : 1, 10 2, 5

(2y 2)(3y 5) (2y 5)(3y 2)

10y , 6y 4y , 15y

Must add up to -11y :

2(2y – 5)(3y + 2)

Factor out GCF : 3

3(3x2 + 7x – 6)

Factors of a : 1, 3

Factors of c : 1, 6 2, 3

(x 2)(3x 3) (x 3)(3x 2)

3x , 6x 2x , 9x

Must add up to 7x :

3(x + 3)(3x – 2)