6.3 Factoring 6.3 Factoring Trinomials IITrinomials II

AxAx22 + bx + c + bx + c

Factoring Trinomials Factoring Trinomials ReviewReview

XX22 + 6x + 5 + 6x + 5 (x )(x )(x )(x )

Find factors of 5 that add to 6: Find factors of 5 that add to 6:

1*6 = 61*6 = 6 1+6 = 71+6 = 7

2*3 = 6 2*3 = 6 2+3 = 52+3 = 5 (x + 2)(x + 3)(x + 2)(x + 3)

Factoring Trinomials Factoring Trinomials where a ≠ 1 where a ≠ 1

Follow these steps:Follow these steps:

1. Find two numbers that multiply to ac 1. Find two numbers that multiply to ac and add to b for axand add to b for ax22 + bx + c + bx + c

2. Replace bx with the sum of the 2 2. Replace bx with the sum of the 2 factors found in step 1. factors found in step 1.

ie: axie: ax22 + bx + c becomes ax + bx + c becomes ax22 + mx + nx + c, + mx + nx + c, where m and n are the factors found in where m and n are the factors found in step 1.step 1.

3. Use grouping to factor this expression 3. Use grouping to factor this expression into 2 binomialsinto 2 binomials

2x2x2 2 + 5x + 2 + 5x + 2

Step 1: ac = 2*2 = 4Step 1: ac = 2*2 = 4 1*4 = 41*4 = 4 1+4 = 51+4 = 5 2*2 = 42*2 = 4 2+2 = 42+2 = 4 m = 1 and n = 4m = 1 and n = 4

Step 2: Rewrite our trinomial by expanding Step 2: Rewrite our trinomial by expanding bxbx

2x2x22 + 1x + 4x + 2 + 1x + 4x + 2Step 3: Group and FactorStep 3: Group and Factor

(2x(2x2 2 + 1x) + (4x + 2)+ 1x) + (4x + 2) x(2x + 1) + 2( 2x + 1)x(2x + 1) + 2( 2x + 1) (2x + 1) (x + 2)(2x + 1) (x + 2)

2x2x2 2 + 5x + 2 + 5x + 2

Questions for thought: Questions for thought:

1.1. Does it matter which order the new Does it matter which order the new factors are entered into the polynomial?factors are entered into the polynomial?

2.2. Do the parenthesis still need to be the Do the parenthesis still need to be the same?same?

3.3. Will signs continue to matter when Will signs continue to matter when finding m and n?finding m and n?

4.4. Does it matter how we group the terms Does it matter how we group the terms for factoring?for factoring?

3z3z22 + z – 2 + z – 2

Step 1: ac = 3*-2 = -6Step 1: ac = 3*-2 = -6 -1*6 = -6-1*6 = -6 -1+6 = 5-1+6 = 5 1* -6 = -6 1* -6 = -6 1+-6 1+-6

= -5= -5 -2*3 = -6-2*3 = -6 -2+3 = 1-2+3 = 1 2* -3 = -6 2* -3 = -6 2+-3 2+-3 = -1= -1

m = -2 and n = 3m = -2 and n = 3Step 2: Rewrite our trinomial by expanding bxStep 2: Rewrite our trinomial by expanding bx

3z3z22 + 3z – 2z – 2 + 3z – 2z – 2 Step 3: Group and FactorStep 3: Group and Factor

(3z(3z2 2 + 3z) + (-2z - 2)+ 3z) + (-2z - 2) 3z(z + 1) - 2( z + 1)3z(z + 1) - 2( z + 1) (z + 1) (3z - 2)(z + 1) (3z - 2)

3z3z22 + z – 2 + z – 2

Step 1: ac = 3*2 = 6Step 1: ac = 3*2 = 6 -1*6 = -6-1*6 = -6 -1+7 = 6-1+7 = 6 1* -6 = -6 1* -6 = -6 1+-7 1+-7

= -6= -6 -2*3 = -6-2*3 = -6 -2+3 = 1-2+3 = 1 2* -3 = -6 2* -3 = -6 2+-3 2+-3 = -1= -1

m = -2 and n = 3m = -2 and n = 3Step 2: Rewrite our trinomial by expanding bxStep 2: Rewrite our trinomial by expanding bx

3z3z22 + 3z – 2z+ 3z – 2z – 2 – 2

Notice that I changed the order of m and n Notice that I changed the order of m and n between step 1 and step 2. Why do you between step 1 and step 2. Why do you think I did this? Do you have to change think I did this? Do you have to change the order to get the correct answer?the order to get the correct answer?

3z3z22 + z – 2 + z – 2

What are the 3 steps for solving this What are the 3 steps for solving this quadratic equation?quadratic equation? Step 1: Multiply a*c. Find the factors Step 1: Multiply a*c. Find the factors

that multiply to that multiply to acac and add to and add to bb Step 2: Expand bx to equal mx + nxStep 2: Expand bx to equal mx + nx Step 3: Group and FactorStep 3: Group and Factor

4x4x33 – 22x – 22x22 + 30x + 30x

Step 0: Factor out the GCF: 2xStep 0: Factor out the GCF: 2x

2x(2x2x(2x22 – 11x + 15) – 11x + 15) Step 1: a*c = 30Step 1: a*c = 30

-1*-30 = 30-1*-30 = 30 -1+-30 = -31-1+-30 = -31

-2*-15 = 30-2*-15 = 30 -2+-15 = -17-2+-15 = -17

-3*-10-3*-10 = 30= 30 -3+-10 = -13-3+-10 = -13

-5*-6 = 30-5*-6 = 30 -5+-6 = -11-5+-6 = -11

4x4x33 – 22x – 22x22 + 30x + 30x

Step 0: Factor out the GCF: 2xStep 0: Factor out the GCF: 2x

2x(2x2x(2x22 – 11x + 15) – 11x + 15) Step 1: a*c = 30Step 1: a*c = 30

-1*-30 = 30-1*-30 = 30 -1+-30 = -31-1+-30 = -31

-2*-15 = 30-2*-15 = 30 -2+-15 = -17-2+-15 = -17

-3*-10-3*-10 = 30= 30 -3+-10 = -13-3+-10 = -13

-5*-6 = 30-5*-6 = 30 -5+-6 = -11-5+-6 = -11

4x4x33 – 22x – 22x22 + 30x + 30x

Step 2: Expand bx to equal mx + nxStep 2: Expand bx to equal mx + nx-11x = -5x + -6x-11x = -5x + -6x

2x(2x2x(2x22 – 5x – 6x + 15) – 5x – 6x + 15) Step 3: Group and FactorStep 3: Group and Factor

2x((2x2x((2x22 – 5x )(– 6x + 15)) – 5x )(– 6x + 15))

2x(x(2x – 5) -3(2x – 5)) 2x(x(2x – 5) -3(2x – 5))

2x(2x – 5) (x – 3) 2x(2x – 5) (x – 3)

4x4x33 – 22x – 22x22 + 30x + 30x

Step 2: Expand bx to equal mx + nxStep 2: Expand bx to equal mx + nx-11x = -5x + -6x-11x = -5x + -6x

2x(2x2x(2x22 – 5x – 6x + 15) – 5x – 6x + 15) Step 3: Group and FactorStep 3: Group and Factor

2x((2x2x((2x22 – 5x )(– 6x + 15)) – 5x )(– 6x + 15))

2x(x(2x – 5) -3(2x – 5)) 2x(x(2x – 5) -3(2x – 5))

Note: The Parenthesis are the SameNote: The Parenthesis are the Same

2x(2x – 5) (x – 3) 2x(2x – 5) (x – 3)

PracticePractice

1. 3x1. 3x22 + 5x + 2 + 5x + 2

2. 6x2. 6x22 + 7x – 3 + 7x – 3

3. 6 + 4y3. 6 + 4y22 – 11y – 11y

PracticePractice

1. 3x1. 3x22 + 5x + 2 + 5x + 2

(3x + 2)(x + 1)(3x + 2)(x + 1)

2. 6x2. 6x22 + 7x – 3 + 7x – 3

(3x – 1)(2x + 3)(3x – 1)(2x + 3)

3. 6 + 4y3. 6 + 4y22 – 11y – 11y

(4y – 3)(y – 2)(4y – 3)(y – 2)

ReviewReview

What is Step 0? When do you need What is Step 0? When do you need to include this step?to include this step?

When will your factors both be When will your factors both be negative?negative?

When will you have one negative and When will you have one negative and one positive factor?one positive factor?

How do you check your answers?How do you check your answers?

??? Questions ?????? Questions ???

6.4 Special Types 6.4 Special Types of Factoringof Factoring

1. Differnce of Squares1. Differnce of Squares

2. Perfect Square Trinomials2. Perfect Square Trinomials

(Sum and Difference of Cubes is not (Sum and Difference of Cubes is not included)included)

Difference of SquaresDifference of Squares

Think back to Chapter 5. What Think back to Chapter 5. What happened when we multiplied a sum happened when we multiplied a sum and difference?and difference?

(a – b)(a + b) = a(a – b)(a + b) = a22 – b – b22

So, the reverse is also true.So, the reverse is also true.

aa22 – b – b22 = (a – b)(a + b) = (a – b)(a + b)

xx22 – 25 – 25

Notice that we do not have a Notice that we do not have a bxbx term. term. This means that we only have the F This means that we only have the F and L in foil; therefore, none of the and L in foil; therefore, none of the procedures from 6.1, 6.2, or 6.3 will procedures from 6.1, 6.2, or 6.3 will work.work.

We need to use aWe need to use a22 – b – b22 = (a – b)(a + b) = (a – b)(a + b)

where a = x and b = 5where a = x and b = 5 XX22 – 25 = (x – 5)(x + 5) – 25 = (x – 5)(x + 5)

xx22 – 36 – 36

We need to use aWe need to use a22 – b – b22 = (a – b)(a + = (a – b)(a + b)b)

where a = x and b = 6where a = x and b = 6 XX22 – 36 = (x – 6)(x + 6) – 36 = (x – 6)(x + 6)

PracticePractice 4x4x22 – 9 – 9

100 – 16t100 – 16t2 2

49y49y22 – 64z – 64z22

PracticePractice 4x4x22 – 9 – 9

a = 2x, b = 3a = 2x, b = 3

(2x – 3) (2x + 3)(2x – 3) (2x + 3) 100 – 16t100 – 16t2 2

a = 10, b = 4ta = 10, b = 4t

(10 – 4t) (10 + 4t)(10 – 4t) (10 + 4t) 49y49y22 – 64z – 64z22

a = 7y, b = 8za = 7y, b = 8z

(7y – 8z) (7y + 8z)(7y – 8z) (7y + 8z)

Perfect Square Perfect Square TrinomialsTrinomials

Think back to Chapter 5. What Think back to Chapter 5. What happened when we squared a happened when we squared a binomial?binomial?

(a + b)(a + b)22 = a = a22 + 2ab + b + 2ab + b22

(a – b)(a – b)22 = a = a22 – 2ab + b – 2ab + b22

So, the reverse is also true.So, the reverse is also true.

aa22 + 2ab + b + 2ab + b22 = (a + b) = (a + b)22

aa22 – 2ab + b – 2ab + b22 = (a – b) = (a – b)22

xx22 + 10x + 25 + 10x + 25

This can be worked 2 different waysThis can be worked 2 different ways The first way is the simplest, but The first way is the simplest, but

depends on whether you recognize the depends on whether you recognize the equation as a perfect square trinomial.equation as a perfect square trinomial.

aa22 + 2ab + b + 2ab + b22 = (a + b) = (a + b)22

Where a = x and b = 5Where a = x and b = 5

xx22 + 10x + 25 = (x + 5) + 10x + 25 = (x + 5)22

xx22 + 10x + 25 + 10x + 25

This can be worked 2 different waysThis can be worked 2 different ways The second way is to use the method we The second way is to use the method we

learned in 6.2learned in 6.2

xx22 + 10x + 25 + 10x + 25

5*5 = 25 and 5+5 = 105*5 = 25 and 5+5 = 10

(x + 5) (x + 5) or (x + 5)(x + 5) (x + 5) or (x + 5)22

4x4x22 - 4x + 1 - 4x + 1

This can be worked 2 different waysThis can be worked 2 different ways The first way is the simplest, but The first way is the simplest, but

depends on whether you recognize the depends on whether you recognize the equation as a perfect square trinomial.equation as a perfect square trinomial.

aa22 + 2ab + b + 2ab + b22 = (a + b) = (a + b)22

Where a = 2x and b = 1Where a = 2x and b = 1

4x4x22 - 4x + 1 = (2x – 1) - 4x + 1 = (2x – 1)22

4x4x22 - 4x + 1 - 4x + 1

This time we need to use the 6.3 This time we need to use the 6.3 methodmethod

4*1 = 44*1 = 4

-2 * -2 = 4 and -2 + -2 = -4 -2 * -2 = 4 and -2 + -2 = -4

(4x(4x22 – 2x) ( – 2x + 1) – 2x) ( – 2x + 1)

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

(2x – 1) (2x – 1) or (2x – 1)(2x – 1) (2x – 1) or (2x – 1)22

PracticePractice xx22 – 4xy + 4y – 4xy + 4y22

9a9a22 – 60a + 100 – 60a + 100

25y25y22 + 20yz + 4z + 20yz + 4z22

PracticePractice xx22 – 4xy + 4y – 4xy + 4y22

a = x, b = 2ya = x, b = 2y

(x – 2y)(x – 2y)22

9a9a22 – 60a + 100 – 60a + 100

a = 3a, b = 10a = 3a, b = 10

(3a – 10) (3a – 10) 25y25y22 + 20yz + 4z + 20yz + 4z22

a = 5y, b = 2za = 5y, b = 2z

(5y + 2z)(5y + 2z)

ReviewReview

What methods can you use to factor What methods can you use to factor a Difference of Squares?a Difference of Squares?

What methods can you use to factor What methods can you use to factor a Perfect Square Trinomial?a Perfect Square Trinomial?

What clues should you look for to What clues should you look for to identify a Difference of Squares?identify a Difference of Squares?

What clues should you look for to What clues should you look for to identify a Perfect Square Trinomial?identify a Perfect Square Trinomial?

??? Questions ?????? Questions ???