Chapters 9.3 and 9.4 Factoring Trinomials

Factoring Trinomials Lesson Objective: Students will know how to use the box method to factor a trinomial

Review Factoring Trinomials Example: Solve (x + 3)(x + 2) Remember we use the box method to solve this problem

Review Factoring Trinomials x x 3x 2x 6 Solve: (x + 3)(x + 2) +3 x x2 * 3 2x * x 2 6 * 3 +2 2

Factoring Trinomials X + 3 +2 x x2 3x 2x 6 x2 + 5x + 6

Factoring Trinomials Today we’re going to learn how to do this in reverse

Factoring Trinomials Example 1: Factor x2 + 7x + 12 We’re going to use the box method to factor this problem

Factoring Trinomials Factor x2 + 7x + 12 Usually we put the problem on the outside, but we were given the answer instead! So we need to find the numbers on the outside

Factoring Trinomials Factor x2 + 7x + 12 In order to find our answer we had to take the numbers from inside the square X2 + 7x + 12

Factoring Trinomials Factor x2 + 7x + 12 Let’s put everything back into the box X2 12 X2 + 7x + 12

Factoring Trinomials Factor x2 + 7x + 12 As you can see, we have one number and 2 spots for it We have to split the 7x into 2 numbers X2 12 X2 + 7x + 12

Factoring Trinomials Factor x2 + 7x + 12 Start by multiplying the 12 and x2 = 12x2 X2 12 X2 + 7x + 12

Factoring Trinomials We’re going to have to set up 2 tables

1x 12x * 2x 6x * 3x 4x * Factoring Trinomials In the first table we put products that multiply to 12x2 Multiplies To 12x2 1x 12x * 2x 6x * 3x 4x *

1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials In the second table we add instead of multiply to get the number in the middle Multiplies To 12x2 Adds To 7x 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x *

1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials Notice the 3x and 4x work for both tables Multiplies To 12x2 Adds To 7x 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x *

1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials Therefore, these are the two numbers that fill in the box Multiplies To 12x2 Adds To 7x 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x *

Factoring Trinomials It doesn’t matter where each one goes, so put them both in the box X2 3x 4x 12 X2 + 7x + 12

x 3 x 4 Factoring Trinomials We can use the Greatest Common Factor to get the numbers on the outside The GCF of x2 and 4x is x The GCF of 3x and 12 is 3 x X2 3x 4 4x 12 X2 + 7x + 12

x 3 x 4 (x + 3) (x + 4) Factoring Trinomials We can then put the numbers on top together for one parenthesis The side is the other parenthesis x X2 3x 4 4x 12 (x + 3) (x + 4) X2 + 7x + 12 =

Let’s try that again! Factoring Trinomials Factor: x2 + 3x – 4 Start with the box

Factoring Trinomials Factor x2 + 3x – 4 Let’s put everything back into the box X2 – 4 X2 + 3x – 4

Factoring Trinomials Factor x2 + 3x + 12 Start by multiplying the -4 and x2 = -4x2 X2 – 4 X2 + 3x – 4

1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x * Factoring Trinomials Set up your two tables Multiplies To -4x2 Adds To 3x 1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x *

1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x * Factoring Trinomials We see that -1x and 4x works for both tables so those are our numbers Multiplies To -4x2 Adds To 3x 1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x *

Factoring Trinomials It doesn’t matter where each one goes, so put them both in the box X2 -1x 4x -4 X2 + 3x – 4

x -1 x 4 Factoring Trinomials We can use the Greatest Common Factor to get the numbers on the outside The GCF of x2 and 4x is x The GCF of -1x and -4 is -1 x X2 -1x 4 4x -4 X2 + 3x – 4

x -1 x 4 Factoring Trinomials Always take the sign closest to the number on the outside! x X2 -1x 4 4x -4 X2 + 3x – 4

x -1 x 4 (x – 1) (x + 4) Factoring Trinomials We can then put the numbers on top together for one parenthesis The side is the other parenthesis x X2 -1x 4 4x -4 (x – 1) (x + 4) X2 + 3x – 4 =

Practice Factoring Trinomials 1. x2 + 8x + 12 2. x2 + 18x + 32 Factor the following: 1. x2 + 8x + 12 2. x2 + 18x + 32 3. x2 – 4x + 4 4. x2 – 7x + 6 5. x2 + 10x + 25

Practice Factoring Trinomials 1. x2 + 8x + 12 2. x2 + 18x + 32 Factor the following: 1. x2 + 8x + 12 2. x2 + 18x + 32 (x + 6)(x + 2) (x + 16)(x + 2) 3. x2 – 4x + 4 4. x2 – 7x + 6 (x – 2)(x – 2) (x – 6)(x – 1) 5. x2 + 10x + 25 (x + 5)(x + 5)

x2 + 6x = 7 -7 -7 x2 + 6x – 7 = 0 Factoring Trinomials Solve: If you see an x2 and an equals sign, you have to get everything on one side of the equation Now we need to factor the left side -7 -7 x2 + 6x – 7 = 0

x2 + 6x – 7 = 0 Factoring Trinomials Let’s put everything back into the box Multiply -7 and x2 = -7x2 X2 – 7 x2 + 6x – 7 = 0

-1x 7x -1x + 7x * Factoring Trinomials Set up your two tables Multiplies To -7x2 Adds To 6x -1x 7x -1x + 7x *

-1x 7x -1x + 7x * Factoring Trinomials We see that -1x and 7x works for both tables so those are our numbers Multiplies To -7x2 Adds To 6x -1x 7x -1x + 7x *

x2 + 6x – 7 = 0 Factoring Trinomials Plug in the two numbers X2 -1x 7x

x -1 x 7 x2 + 6x – 7 = 0 Factoring Trinomials Find the GCF to put on the outside of the box x X2 -1x 7 7x – 7 x2 + 6x – 7 = 0

x -1 x 7 (x – 1) (x + 7) x2 + 6x – 7 = 0 Factoring Trinomials Replace the equation with your answer x X2 -1x 7 7x – 7 (x – 1) (x + 7) x2 + 6x – 7 = 0

(x – 1) (x + 7) = 0 x + 7 = 0 x – 1 = 0 -7 -7 +1 +1 x = -7 x = 1 Factoring Trinomials (x – 1) (x + 7) = 0 x + 7 = 0 x – 1 = 0 -7 -7 +1 +1 x = -7 x = 1 Just a reminder: x*y = 0 means that either x or y has to be zero! We must set both parenthesis equal to zero and solve

Practice Factoring Trinomials 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 Factor the following: 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0 5. x2 + 10x – 24 = 0

Practice Factoring Trinomials 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 Factor the following: 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 x = -3 and -4 x = -8 or -2 3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0 x = 2 or 3 x = 6 or -1 5. x2 + 10x – 24 = 0 x = -12 or 2

2x2 + 15x + 18 Factoring Trinomials Example 4: Factor We’re going to work this like the other problems

2x2 + 15x + 18 Factoring Trinomials Start with the box! Multiply 18 and 2x2 = 36x2 2x2 18 2x2 + 15x + 18

Factoring Trinomials Set up your two tables Multiplies To 36x2 Adds To 15x 1x 36x 1x + 36x * 2x 18x 2x 18x + * 3x 12x 3x 12x + *

Factoring Trinomials 3x and 4x works for both, so those are our numbers Multiplies To 36x2 Adds To 15x 1x 36x 1x + 36x * 2x 18x 2x 18x + * 3x 12x 3x 12x + *

2x2 + 15x + 18 Factoring Trinomials Plug in the two numbers 2x2 12x 3x

x 6 2x 3 2x2 + 15x + 18 Factoring Trinomials Find the GCF to put on the outside of the box 2x 2x2 12x 3 3x 18 2x2 + 15x + 18

x 6 2x 3 2x2 + 15x + 18 (x + 6) (2x + 3) Factoring Trinomials We can then put the numbers on top together for one parenthesis The side is the other parenthesis 2x 2x2 12x 3 3x 18 2x2 + 15x + 18 (x + 6) (2x + 3)

2x2 + 3x – 6 2x2 + 3x – 6 Factoring Trinomials Example 5: Factor: Plug them into the box Multiply -6 and 2x2 = -12x2 2x2 + 3x – 6 2x2 -6 2x2 + 3x – 6

-1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + * Factoring Trinomials Set up your two tables Multiplies To -12x2 Adds To 3x -1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + *

-1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + * Factoring Trinomials No factors work, so we can’t factor this equation Multiplies To -12x2 Adds To 3x -1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + *

2x2 + 3x – 6 Prime Factoring Trinomials Since we can’t factor this problem we call it Prime

Practice Factoring Trinomials 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2 Factor the following: 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2 3. 4x2 + 8x – 5 4. 4x2 – 3x – 3 5. 6x2 – 13x + 6

Practice Factoring Trinomials 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2 Factor the following: 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2 (x + 2)(2x + 1) (3x – 1)(x – 2) 3. 4x2 + 8x – 5 4. 4x2 – 3x – 3 (2x + 5)(2x – 1) Prime 5. 6x2 – 13x + 6 (3x – 2)(2x – 3)

12x2 – 32x – 12 Factoring Trinomials Example 6: Factor The first thing we should do is look for a common factor This equation has a common factor The GCF is 4 12x2 – 32x – 12

12x2 – 32x – 12 ___ ___ __ 4 4 4 (3x2 – 8x – 3) 4 Factoring Trinomials Example 5: Factor Factor out the 4 12x2 – 32x – 12 ___ ___ __ 4 4 4 (3x2 – 8x – 3) 4

4(3x2 – 8x – 3) Factoring Trinomials Factor what’s inside the parenthesis, ignore the 4 Plug into the box Multiply -3 and 3x2 = -9x2 3x2 -3 4(3x2 – 8x – 3)

1x -9x 1x + -9x * Factoring Trinomials Set up your two tables Multiplies To -9x2 Adds To -8x 1x -9x 1x + -9x *

1x -9x 1x + -9x * Factoring Trinomials 1x and -9x works for both, so those are our numbers Multiplies To -9x2 Adds To -8x 1x -9x 1x + -9x *

4(3x2 – 8x – 3) Factoring Trinomials Plug in the two numbers 3x2 1x -3 4(3x2 – 8x – 3)

3x 1 x -3 4(3x2 – 8x – 3) Factoring Trinomials Find the GCF to put on the outside of the box x 3x2 1x -3 -9x -3 4(3x2 – 8x – 3)

3x 1 x -3 4(3x2 – 8x – 3) (3x + 1) (x – 3) 4 Factoring Trinomials Find the GCF to put on the outside of the box Put the 4 back in front x 3x2 1x -3 -9x -3 4(3x2 – 8x – 3) (3x + 1) (x – 3) 4

Practice Factoring Trinomials 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6 Factor the following: 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6 3. 20x2 + 40x – 25 4. 18x3 + 15x2 – 18x 5. 36x3 – 78x2 + 36x

Practice Factoring Trinomials 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6 Factor the following: 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6 2(x + 2)(2x + 1) 3(3x – 1)(x – 2) 3. 20x2 + 40x – 25 4. 18x3 + 15x2 – 18x 5(2x + 5)(2x – 1) 3x(2x+3)(3x-2) 5. 36x3 – 78x2 + 36x 6x(3x – 2)(2x – 3)