8.6: FACTORING ax2 + bx + c where a ≠ 1

Factoring: A process used to break down any polynomial into simpler polynomials.

FACTORING ax2 + bx + c Procedure:

1) Always look for the GCF of all the terms

2) Factor the remaining terms – pay close attention to the value of coefficient a and follow the proper steps.

3) Re-write the original polynomial as a product of the polynomials that cannot be factored any further.

GOAL:

FACTORING: When AC is POSSIBLE.

Ex: What is the FACTORED form of:

5x2+11x+2?

FACTORING: To factor a quadratic trinomial with a coefficient ≠ 1 in the x2, we must look at the b and ac coefficients:

5x2+11x+2 ax2+bx+c b= +11 ac =(5)(2) Look at the factors of ac: ac = +10 : (1)(10), (2)(5)

Take the pair that equals to b when adding the two integers.In our case it is 1x10 since 1+10 =11= b

Re-write using factors of ac that = b.

5x2+11x+2 5x2 + 1x + 10x + 2 Look at the GCF of the first two terms:

Thus the factored form is: (5x+1) (x+2)

5x2 + 1x x(5x + 1) Look at the GCF of the last two terms: 10x + 2 2(5x + 1)

Look at the GCF of both: x(5x + 1) + 2(5x + 1)

YOU TRY IT:

Ex: What is the FACTORED form of:

6x2+13x+5?

SOLUTION: To factor a quadratic trinomial with a coefficient ≠ 1 in the x2, we must look at the b and ac coefficients:

6x2+13x+5 ax2+bx+c b= +13 ac =(6)(5) Look at the factors of C: ac = +30 :(1)(30), (2)(15), (3)(10)Take the pair that equals to b when adding the two integers.In our case it is 3x10 since 3+10 =13= b

Re-write using factors of ac that = b.

6x2+13x+5 6x2 + 3x + 10x + 5 Look at the GCF of the first two terms:

Thus the factored form is:(3x+5)(2x+1)

6x2 + 3x 3x(2x + 1) Look at the GCF of the last two terms: 10x + 5 5(2x + 1)

Look at the GCF of both: 3x(2x + 1)+ 5(2x + 1)

CLASSWORK:

Page 508-509:

Problems: 1, 2, 5, 8, 9, 11, 20.

FACTORING: When ac is Negative.

Ex: What is the FACTORED form of:

3x2+4x-15?

FACTORING: Since a ≠ 1, we still look at the b and ac coefficients: 3x2+4x-15 ax2+bx+c b= +4 ac =(3)(-15) Look at the factors of ac: ac = -45 : (-1)(45), (1)(-45)

(-3)(15), (3)(-15) (-5)(9), (5)(-9)

Take the pair that equals to b when adding the two integers.

In our case it is (-5)(9)since -5+9 =+4 =b

Re-write: using factors of ac that = b.

3x2+4x-15 3x2 -5x + 9x -15 Look at the GCF of the first two terms:

Thus the factored form is: (3x-5) (x+3)

3x2 - 5x x(3x - 5) Look at the GCF of the last two terms: 9x -15 3(3x -5)

Look at the GCF of both: x(3x - 5) + 3(3x - 5)

YOU TRY IT:

Ex: What is the FACTORED form of:

10x2+31x-14?

FACTORING: Since a ≠ 1, we still look at the b and ac coefficients: 10x2+31x-14 ax2+bx+c b= +31 ac =(10)(-14) Look at the factors of ac: ac = -140 : (-1)(140), (1)(-140)

(-2)(70), (2)(-70) (-4)(35), (4)(-35)

Take the pair that equals to b when adding the two integers.

This time it is(-4)(35)since -4+35=+31=b

Re-write using factors of ac that = b.

10x2+31x-14 10x2-4x + 35x -14 Look at the GCF of the first two terms:

Thus the factored form is:(2x+7)(5x-2)

10x2 - 4x 2x(5x - 2) Look at the GCF of the last two terms: 35x -14 7(5x - 2)

Look at the GCF of both: 2x(5x - 2)+ 7(5x - 2)

CLASSWORK:

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Problems: 3, 6, 14, 16, 17, 18, 34.

REAL-WORLD:

The area of a rectangular knitted blanket is

15x2-14x-8.What are the possible dimensions of the blanket?

SOLUTION: Since a ≠ 1, we still look at the b and ac coefficients: 15x2-14x-8 ax2+bx+c b= -14 ac =(15)(-8) Look at the factors of ac: ac = -120 : (-1)(120), (1)(-120) (-2)(60), (2)(-60),(-3)(40), (3)(-40) (-4)(30), (4)(-30), (-5)(24), (5)(-24) (-6)(20), (6)(-20), (-8)(15), (8)(-15)

This time it is(6)(-20)since 6-20=-14=b

Re-write using factors of ac that = b.

15x2-14x-8 15x2 + 6x - 20x -8 Look at the GCF of the first two terms:

Thus the factored form is:(3x-4)(5x+2)

15x2 + 6x 3x(5x + 2) Look at the GCF of the last two terms: -20x -8 -4 (5x + 2)

Look at the GCF of both: 3x(5x + 2)- 4(5x + 2)

VIDEOS: FactoringQuadratics

Factoring when a ≠ 1:

http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_factoring/v/factoring-trinomials-by-grouping-5

http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_factoring/v/factoring-trinomials-by-grouping-6

CLASSWORK:

Page 508-509:

Problems: 4, 7, 15, 21, 25, 36, 37.