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

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factoring trinomials
Factoring Trinomials
  • Lesson Objective:
  • Students will know how to use the box method to factor a trinomial
factoring trinomials1
Factoring Trinomials

Review

  • Example: Solve (x + 3)(x + 2)
  • Remember we use the box method to solve this problem
factoring trinomials2
Factoring Trinomials

Review

  • Solve: (x + 3)(x + 2)

x

+3

x

x

x2

* x

x

3x

* 3

6

2x

* x

2

* 3

+2

2

factoring trinomials3
Factoring Trinomials

X + 3

+2 x

x2

3x

2x

6

x2

+ 5x

+ 6

factoring trinomials4
Factoring Trinomials
  • Today we’re going to learn how to do this in reverse
factoring trinomials5
Factoring Trinomials
  • Example 1: Factor x2 + 7x + 12
  • We’re going to use the box method to factor this problem
factoring trinomials6
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 trinomials7
Factoring Trinomials
  • Factor x2 + 7x + 12
  • In order to find our answer we had to take the numbers from inside the square

+ 7x

+ 12

X2

factoring trinomials8
Factoring Trinomials
  • Factor x2 + 7x + 12
  • Let’s put everything back into the box

X2

12

+ 7x

+ 12

X2

factoring trinomials9
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

+ 7x

+ 12

X2

factoring trinomials10
Factoring Trinomials
  • Factor x2 + 7x + 12
  • Start by multiplying the 12 and x2
  • = 12x2

X2

12

+ 7x

+ 12

X2

factoring trinomials11
Factoring Trinomials
  • We’re going to have to set up 2 tables
factoring trinomials12
Factoring Trinomials
  • In the first table we put products that multiply to 12x2

1x

12x

*

6x

2x

*

3x

4x

*

factoring trinomials13
Factoring Trinomials
  • In the second table we add instead of multiply to get the number in the middle

1x

12x

1x

+

12x

*

+

6x

2x

6x

2x

*

+

3x

4x

3x

4x

*

factoring trinomials14
Factoring Trinomials
  • Notice the 3x and 4x work for both tables

1x

12x

1x

+

12x

*

+

6x

2x

6x

2x

*

+

3x

4x

3x

4x

*

factoring trinomials15
Factoring Trinomials
  • Therefore, these are the two numbers that fill in the box

1x

12x

1x

+

12x

*

+

6x

2x

6x

2x

*

+

3x

4x

3x

4x

*

factoring trinomials16
Factoring Trinomials
  • It doesn’t matter where each one goes, so put them both in the box

X2

3x

4x

12

+ 7x

+ 12

X2

factoring trinomials17
Factoring Trinomials

x

3

  • 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

+ 7x

+ 12

X2

factoring trinomials18
Factoring Trinomials

x

3

  • 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 + 4)

(x + 3)

+ 7x

+ 12 =

X2

factoring trinomials19
Factoring Trinomials

Let’s try that again!

  • Factor: x2 + 3x – 4
  • Start with the box
factoring trinomials20
Factoring Trinomials
  • Factor x2 + 3x – 4
  • Let’s put everything back into the box

X2

– 4

+ 3x

– 4

X2

factoring trinomials21
Factoring Trinomials
  • Factor x2 + 3x + 12
  • Start by multiplying the -4 and x2
  • = -4x2

X2

– 4

+ 3x

– 4

X2

factoring trinomials22
Factoring Trinomials
  • Set up your two tables

1x

-4x

1x

+

-4x

*

+

4x

-1x

4x

-1x

*

+

2x

-2x

2x

-2x

*

factoring trinomials23
Factoring Trinomials
  • We see that -1x and 4x works for both tables so those are our numbers

1x

-4x

1x

+

-4x

*

+

4x

-1x

4x

-1x

*

+

2x

-2x

2x

-2x

*

factoring trinomials24
Factoring Trinomials
  • It doesn’t matter where each one goes, so put them both in the box

X2

-1x

4x

-4

+ 3x

– 4

X2

factoring trinomials25
Factoring Trinomials

x

-1

  • 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

+ 3x

– 4

X2

factoring trinomials26
Factoring Trinomials

x

-1

  • Always take the sign closest to the number on the outside!

x

X2

-1x

4

4x

-4

+ 3x

– 4

X2

factoring trinomials27
Factoring Trinomials

x

-1

  • 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 + 4)

(x – 1)

+ 3x

– 4 =

X2

factoring trinomials28
Factoring Trinomials

Practice

  • Factor the following:

1. x2 + 8x + 12

2. x2 + 18x + 32

3. x2 – 4x + 4

4. x2 – 7x + 6

5. x2 + 10x + 25

factoring trinomials29
Factoring Trinomials

Practice

  • 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 – 6)(x – 1)

(x – 2)(x – 2)

5. x2 + 10x + 25

(x + 5)(x + 5)

factoring trinomials30
Factoring Trinomials

x2 + 6x = 7

  • 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

factoring trinomials31
Factoring Trinomials
  • Let’s put everything back into the box
  • Multiply -7 and x2
  • = -7x2

X2

– 7

x2 + 6x – 7 = 0

factoring trinomials32
Factoring Trinomials
  • Set up your two tables

-1x

7x

-1x

+

7x

*

factoring trinomials33
Factoring Trinomials
  • We see that -1x and 7x works for both tables so those are our numbers

-1x

7x

-1x

+

7x

*

factoring trinomials34
Factoring Trinomials
  • Plug in the two numbers

X2

-1x

– 7

7x

x2 + 6x – 7 = 0

factoring trinomials35
Factoring Trinomials

x

-1

  • Find the GCF to put on the outside of the box

x

X2

-1x

7

– 7

7x

x2 + 6x – 7

= 0

factoring trinomials36
Factoring Trinomials

x

-1

  • Replace the equation with your answer

x

X2

-1x

7

– 7

7x

(x + 7)

(x – 1)

x2 + 6x – 7

= 0

factoring trinomials37
Factoring Trinomials

(x + 7)

(x – 1)

= 0

  • 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

x + 7

= 0

x – 1

= 0

-7-7

+1+1

x = -7

x = 1

factoring trinomials38
Factoring Trinomials

Practice

  • 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

factoring trinomials39
Factoring Trinomials

Practice

  • 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 = 6 or -1

x = 2 or 3

5. x2 + 10x – 24 = 0

x = -12 or 2

factoring trinomials40
Factoring Trinomials

2x2 + 15x + 18

  • Example 4: Factor
  • We’re going to work this like the other problems
factoring trinomials41
Factoring Trinomials
  • Start with the box!
  • Multiply 18 and 2x2
  • = 36x2

2x2

18

2x2 + 15x + 18

factoring trinomials42
Factoring Trinomials
  • Set up your two tables

1x

36x

1x

36x

+

*

2x

18x

2x

18x

+

*

3x

12x

3x

12x

+

*

factoring trinomials43
Factoring Trinomials
  • 3x and 4x works for both, so those are our numbers

1x

36x

1x

+

36x

*

2x

18x

2x

18x

+

*

3x

12x

3x

12x

+

*

factoring trinomials44
Factoring Trinomials
  • Plug in the two numbers

2x2

12x

18

3x

2x2 + 15x + 18

factoring trinomials45
Factoring Trinomials

x

6

  • Find the GCF to put on the outside of the box

2x

2x2

12x

3

18

3x

2x2 + 15x + 18

factoring trinomials46
Factoring Trinomials

x

6

  • We can then put the numbers on top together for one parenthesis
  • The side is the other parenthesis

2x

2x2

12x

3

18

3x

2x2 + 15x + 18

(2x + 3)

(x + 6)

factoring trinomials47
Factoring Trinomials
  • Example 5: Factor:
  • Plug them into the box
  • Multiply -6 and 2x2
  • = -12x2

2x2 + 3x – 6

2x2

-6

2x2 + 3x – 6

factoring trinomials48
Factoring Trinomials
  • Set up your two tables

-1x

12x

-1x

+

12x

*

-2x

6x

-2x

6x

+

*

-3x

4x

-3x

4x

+

*

factoring trinomials49
Factoring Trinomials
  • No factors work, so we can’t factor this equation

-1x

12x

-1x

+

12x

*

-2x

6x

-2x

6x

+

*

-3x

4x

-3x

4x

+

*

factoring trinomials50
Factoring Trinomials
  • Since we can’t factor this problem we call it

2x2 + 3x – 6

Prime

factoring trinomials51
Factoring Trinomials

Practice

  • Factor the following:

1. 2x2 + 5x + 2

2. 3x2 – 7x + 2

3. 4x2 + 8x – 5

4. 4x2 – 3x – 3

5. 6x2 – 13x + 6

factoring trinomials52
Factoring Trinomials

Practice

  • 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

Prime

(2x + 5)(2x – 1)

5. 6x2 – 13x + 6

(3x – 2)(2x – 3)

factoring trinomials53
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

factoring trinomials54
Factoring Trinomials
  • Example 5: Factor
  • Factor out the 4

12x2 – 32x – 12

___ ___ __

4 4 4

– 3)

(3x2

– 8x

4

factoring trinomials55
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)

factoring trinomials56
Factoring Trinomials
  • Set up your two tables

1x

-9x

1x

+

-9x

*

factoring trinomials57
Factoring Trinomials
  • 1x and -9x works for both, so those are our numbers

1x

-9x

1x

+

-9x

*

factoring trinomials58
Factoring Trinomials
  • Plug in the two numbers

3x2

1x

-3

-9x

4(3x2 – 8x – 3)

factoring trinomials59
Factoring Trinomials

3x

1

  • Find the GCF to put on the outside of the box

x

3x2

1x

-3

-3

-9x

4(3x2 – 8x – 3)

factoring trinomials60
Factoring Trinomials

3x

1

  • Find the GCF to put on the outside of the box
  • Put the 4 back in front

x

3x2

1x

-3

-3

-9x

4(3x2 – 8x – 3)

(3x + 1)

(x – 3)

4

factoring trinomials61
Factoring Trinomials

Practice

  • Factor the following:

1. 4x2 + 10x + 4

2. 9x2 – 21x + 6

3. 20x2 + 40x – 25

4. 18x3 + 15x2 – 18x

5. 36x3 – 78x2 + 36x

factoring trinomials62
Factoring Trinomials

Practice

  • 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

3x(2x+3)(3x-2)

5(2x + 5)(2x – 1)

5. 36x3 – 78x2 + 36x

6x(3x – 2)(2x – 3)