Factoring Trinomials
Lesson Objective:
Students will know how to use the box method to factor a trinomial
Factoring Trinomials
Example: Solve (x + 3)(x + 2)
Remember we use the box method to solve this problem
Review
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
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
Factoring Trinomials
Factor x2 + 7x + 12
Start by multiplying the 12 and x2
= 12x2
X2 + 7x + 12
X2
12
Factoring Trinomials
In the first table we put products that multiply to 12x2
Multiplies
To 12x2
1x * 12x
2x
3x*
*
6x
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
3x*
*
6x
4x
2x
3x
+
+
6x
4x
Factoring Trinomials
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
Factoring Trinomials
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
Factoring Trinomials
It doesn’t matter where each one goes, so put them both in the box
X2 + 7x + 12
X2
124x
3x
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
X2 + 7x + 12
X2
124x
3x
x 3
x
4
Factoring Trinomials
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)
Factoring Trinomials
Factor x2 + 3x + 12
Start by multiplying the -4 and x2
= -4x2
X2 + 3x
X2
– 4
– 4
Factoring Trinomials
Set up your two tables
Multiplies
To -4x2 Adds To 3x
1x * -4x 1x + -4x
-1x
2x*
*
4x
-2x
-1x
2x
+
+
4x
-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
2x*
*
4x
-2x
-1x
2x
+
+
4x
-2x
Factoring Trinomials
It doesn’t matter where each one goes, so put them both in the box
X2 + 3x – 4
X2
-44x
-1x
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
X2 + 3x
X2
4x
-1x
x -1
x
4
– 4
-4
Factoring Trinomials
Always take the sign closest to the number on the outside!
X2 + 3x
X2
4x
-1x
x -1
x
4
– 4
-4
Factoring Trinomials
We can then put the numbers on top together for one parenthesis
The side is the other parenthesis
X2 + 3x – 4 =
X2
-44x
-1x
x -1
x
4
(x – 1) (x + 4)
Factoring Trinomials
Factor the following:
1. x2 + 8x + 12
3. x2 – 4x + 4 4. x2 – 7x + 6
5. x2 + 10x + 25
2. x2 + 18x + 32
PRACTICE
Factoring Trinomials
Factor the following:
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)
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
X2 + 6X = 7
X2 + 6X – 7 = 0-7 -7
Factoring Trinomials
Let’s put everything back into the box
Multiply -7 and x2
= -7x2
X2
– 7
X2 + 6X – 7 = 0
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
Factoring Trinomials
Find the GCF to put on the outside of the box
X2
x -1
x
7 – 7 7x
-1x
X2 + 6X – 7 = 0
Factoring Trinomials
Replace the equation with your answer
X2
x -1
x
7 – 7 7x
-1x
X2 + 6X – 7 = 0(x – 1) (x + 7)
Factoring Trinomials
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
Factoring Trinomials
Factor the following:
1. x2 + 7x + 12 = 0
3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0
5. x2 + 10x – 24 = 0
2. x2 + 10x = -16
PRACTICE
Factoring Trinomials
Factor the following:
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
Factoring Trinomials
Example 4: Factor
We’re going to work this like the other problems
2x2 + 15x + 18
Factoring Trinomials
Set up your two tables
Multiplies
To 36x2 Adds To 15x
1x * 36x +
2x * 18x +3x * 12x +
1x 36x
2x 18x
3x 12x
Factoring Trinomials
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
Factoring Trinomials
Find the GCF to put on the outside of the box
2x2
x 6
2x
3 183x
12x
2x2 + 15x + 18
Factoring Trinomials
We can then put the numbers on top together for one parenthesis
The side is the other parenthesis
2x2
x 6
2x
3 183x
12x
(x + 6)(2x + 3)2x2 + 15x + 18
Factoring Trinomials
Example 5: Factor:
Plug them into the box
Multiply -6 and 2x2
= -12x2
2x2
-6
2x2 + 3x – 6
2x2 + 3x – 6
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
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
Factoring Trinomials
Factor the following:
1. 2x2 + 5x + 2
3. 4x2 + 8x – 5 4. 4x2 – 3x – 3
5. 6x2 – 13x + 6
2. 3x2 – 7x + 2
PRACTICE
Factoring Trinomials
Factor the following:
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
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 Trinomials
Example 5: Factor
Factor out the 4
12x2 – 32x – 12___ ___ __4 4 4
(3x24 – 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)
Factoring Trinomials
1x and -9x works for both, so those are our numbers
Multiplies
To -9x2 Adds To -8x
1x * -9x 1x + -9x
Factoring Trinomials
Find the GCF to put on the outside of the box
3x2
3x 1
x
-3 -3-9x
1x
4(3x2 – 8x – 3)
Factoring Trinomials
Find the GCF to put on the outside of the box
Put the 4 back in front
3x2
-3-9x
1x
(3x + 1)(x – 3)
3x 1
x
-34(3x2 – 8x – 3)
4
Factoring Trinomials
Factor the following:
1. 4x2 + 10x + 4
3. 20x2 + 40x – 25 4. 18x3 + 15x2 – 18x
5. 36x3 – 78x2 + 36x
2. 9x2 – 21x + 6
PRACTICE