Chapters 9.3 and 9.4. Factoring Trinomials Lesson Objective: Students will know how to use the box method to factor a trinomial

FACTORING TRINOMIALS

Chapters 9.3 and 9.4

Factoring Trinomials

Lesson Objective:

Students will know how to use the box method to factor a trinomial


Example: Solve (x + 3)(x + 2)

Remember we use the box method to solve this problem

Review


Solve: (x + 3)(x + 2)x +3

x

+2

x * xx2 x * 33x

2 * x2x 2 * 36

Review


X + 3

+

2

x x2 3x

2x 6

x2 + 5x + 6


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


Example 1: Factor x2 + 7x + 12

We’re going to use the box method to factor this problem


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


Factor x2 + 7x + 12

In order to find our answer we had to take the numbers from inside the square

X2 + 7x + 12


Factor x2 + 7x + 12

Let’s put everything back into the box

X2 + 7x + 12

X2

12


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 + 7x + 12

X2

12


Factor x2 + 7x + 12

Start by multiplying the 12 and x2

= 12x2

X2 + 7x + 12

X2

12


We’re going to have to set up 2 tables


In the first table we put products that multiply to 12x2

Multiplies

To 12x2

1x * 12x

2x

3x*

*

6x

4x


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

3x*

*

6x

4x

2x

3x

+

+

6x

4x


Notice the 3x and 4x work for both tables

Multiplies

To 12x2 Adds To 7x

1x * 12x 1x + 12x

2x

3x*

*

6x

4x

2x

3x

+

+

6x

4x


Therefore, these are the two numbers that fill in the box

Multiplies

To 12x2 Adds To 7x

1x * 12x 1x + 12x

2x

3x*

*

6x

4x

2x

3x

+

+

6x

4x


It doesn’t matter where each one goes, so put them both in the box

X2 + 7x + 12

X2

124x

3x


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

X2 + 7x + 12

X2

124x

3x

x 3

x

4


We can then put the numbers on top together for one parenthesis

The side is the other parenthesis

X2 + 7x + 12 =

X2

124x

3x

x 3

x

4

(x + 3)(x + 4)


Factor: x2 + 3x – 4

Start with the box

Let’s try that again!


Factor x2 + 3x – 4


X2 + 3x – 4

X2

– 4


Factor x2 + 3x + 12

Start by multiplying the -4 and x2

= -4x2

X2 + 3x

X2

– 4

– 4


Set up your two tables

Multiplies

To -4x2 Adds To 3x

1x * -4x 1x + -4x

-1x

2x*

*

4x

-2x

-1x

2x

+

+

4x

-2x


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

2x*

*

4x

-2x

-1x

2x

+

+

4x

-2x


It doesn’t matter where each one goes, so put them both in the box

X2 + 3x – 4

X2

-44x

-1x


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

X2 + 3x

X2

4x

-1x

x -1

x

4

– 4

-4


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

X2 + 3x

X2

4x

-1x

x -1

x

4

– 4

-4




X2 + 3x – 4 =

X2

-44x

-1x

x -1

x

4

(x – 1) (x + 4)


Factor the following:

1. x2 + 8x + 12

3. x2 – 4x + 4 4. x2 – 7x + 6

5. x2 + 10x + 25

2. x2 + 18x + 32

PRACTICE



1. x2 + 8x + 12

3. x2 – 4x + 4 4. x2 – 7x + 6

5. x2 + 10x + 25

2. x2 + 18x + 32

PRACTICE

(x + 6)(x + 2)

(x – 2)(x – 2)

(x + 5)(x + 5)

(x + 16)(x + 2)

(x – 6)(x – 1)


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

X2 + 6X = 7

X2 + 6X – 7 = 0-7 -7



Multiply -7 and x2

= -7x2

X2

– 7

X2 + 6X – 7 = 0



Multiplies

To -7x2 Adds To 6x

-1x * 7x -1x + 7x


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


Plug in the two numbers

X2

– 7

X2 + 6X – 7 = 0

7x

-1x


Find the GCF to put on the outside of the box

X2

x -1

x

7 – 7 7x

-1x

X2 + 6X – 7 = 0


Replace the equation with your answer

X2

x -1

x

7 – 7 7x

-1x

X2 + 6X – 7 = 0(x – 1) (x + 7)


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

= 0(x – 1) (x + 7)x – 1 = 0= 0 x + 7

+1 +1x = 1

-7 -7x = -7



1. x2 + 7x + 12 = 0

3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0

5. x2 + 10x – 24 = 0

2. x2 + 10x = -16

PRACTICE



1. x2 + 7x + 12 = 0

3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0

5. x2 + 10x – 24 = 0

2. x2 + 10x = -16

PRACTICE

x = -3 and -4

x = 2 or 3

x = -12 or 2

x = -8 or -2

x = 6 or -1


Example 4: Factor

We’re going to work this like the other problems

2x2 + 15x + 18


Start with the box!

Multiply 18 and 2x2

= 36x2

2x2

18

2x2 + 15x + 18



Multiplies

To 36x2 Adds To 15x

1x * 36x +

2x * 18x +3x * 12x +

1x 36x

2x 18x

3x 12x


3x and 4x works for both, so those are our numbers

Multiplies

To 36x2 Adds To 15x

* +

* +

* +

1x 36x

2x 18x

3x 12x

1x 36x

2x 18x

3x 12x



2x2

183x

12x

2x2 + 15x + 18



2x2

x 6

2x

3 183x

12x

2x2 + 15x + 18




2x2

x 6

2x

3 183x

12x

(x + 6)(2x + 3)2x2 + 15x + 18


Example 5: Factor:

Plug them into the box

Multiply -6 and 2x2

= -12x2

2x2

-6

2x2 + 3x – 6

2x2 + 3x – 6



Multiplies

To -12x2 Adds To 3x

-1x * 12x -1x + 12x

-2x * 6x -2x + 6x

-3x * 4x -3x + 4x


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


Since we can’t factor this problem we call it

2x2 + 3x – 6

Prime



1. 2x2 + 5x + 2

3. 4x2 + 8x – 5 4. 4x2 – 3x – 3

5. 6x2 – 13x + 6

2. 3x2 – 7x + 2

PRACTICE



1. 2x2 + 5x + 2

3. 4x2 + 8x – 5 4. 4x2 – 3x – 3

5. 6x2 – 13x + 6

2. 3x2 – 7x + 2

PRACTICE

(x + 2)(2x + 1)

(2x + 5)(2x – 1)

(3x – 2)(2x – 3)

(3x – 1)(x – 2)

Prime


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


Example 5: Factor

Factor out the 4

12x2 – 32x – 12___ ___ __4 4 4

(3x24 – 8x – 3)


Factor what’s inside the parenthesis, ignore the 4

Plug into the box

Multiply -3 and 3x2

= -9x2

3x2

-3

4(3x2 – 8x – 3)



Multiplies

To -9x2 Adds To -8x

1x * -9x 1x + -9x


1x and -9x works for both, so those are our numbers

Multiplies

To -9x2 Adds To -8x

1x * -9x 1x + -9x



3x2

-3-9x

1x

4(3x2 – 8x – 3)



3x2

3x 1

x

-3 -3-9x

1x

4(3x2 – 8x – 3)



Put the 4 back in front

3x2

-3-9x

1x

(3x + 1)(x – 3)

3x 1

x

-34(3x2 – 8x – 3)

4



1. 4x2 + 10x + 4

3. 20x2 + 40x – 25 4. 18x3 + 15x2 – 18x

5. 36x3 – 78x2 + 36x

2. 9x2 – 21x + 6

PRACTICE



1. 4x2 + 10x + 4

3. 20x2 + 40x – 25 4. 18x3 + 15x2 – 18x

5. 36x3 – 78x2 + 36x

2. 9x2 – 21x + 6

PRACTICE

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

5(2x + 5)(2x – 1)

6x(3x – 2)(2x – 3)

3(3x – 1)(x – 2)

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

Documents

Chapters 9.3 and 9.4. Factoring Trinomials Lesson Objective: Students will know how to use the box method to factor a trinomial