Transcript Factor Trinomials by Grouping

Objective
The student will be able to:
factor trinomials with grouping.
Designed by Skip Tyler, Varina High School
Review: (y + 2)(y + 4)
y2
+4y
+2y
+8
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 6y + 8
In this lesson, we will begin with y2 + 6y + 8 as our
problem and finish with (y + 2)(y + 4) as our answer.
1) Factor y2 + 6y + 8
Product of the first and
last coefficients
Middle
coefficient
The goal is to find two factors in the first column that
add up to the middle term in the second column.
We’ll work it out in the next few slides.
1) Factor
2
y
M
A
+ 6y + 8
Create your table.
Product of the
first and last
coefficients
Multiply
+8
Add
+6
Middle
coefficient
Here’s your task…
What numbers multiply to +8 and add to +6?
If you cannot figure it out right away, write
the combinations.
1) Factor
2
y
+ 6y + 8
Place the factors in the table.
Multiply
+8
Which has
a sum
of +6?
Add
+6
+1, +8 +9, NO
+2, +4 +6, YES!!
We are going to use these numbers in the next step!
1) Factor y2 + 6y + 8
Multiply
+8
Add
+6
+2, +4 +6, YES!!
Next we will write two sets for our factors.
(x + 2)(x + 6)
y2 + 6y + 8 = (x + 2)(x + 6)
We can check using FOIL.
NOTE: We can only use this method for factoring when
the first term has a coefficient of 1.
1) Here is the method using grouping, we will use
this tomorrow.
We use our factors to rewrite the expression like this:
y2 + 6y + 8
y2 + 2y + 4y + 8
Now we have two groups
y2 + 2y + 4y + 8
Find the GCF of each group and factor it out.
y(y + 2) +4(y + 2)
If things are done
Factor out the
GCF’s. Write them
in their own group.
right, the parentheses
should be the same.
(y + 4)(y + 2)
This method works with all trinomials that can be factored.
2) Factor y2 + 7y + 12
The goal is to find two factors in the first column that add up to the middle term
in the second column.
Product of the first and last coefficients
+1, +12
+2, +6
+3, +4
Middle coefficient
+13, NO
+8, NO
+7, YES!!
Now we will make our two sets for the
factors.
(y + 3)(y + 4)
y2 + 7y + 12 = (y + 3)(y + 4)
Check by multiplying with FOIL.
Factor
2
x
+ 13x + 30
Factor
2
x
+ 3x + 2
M
A
3) Factor x2 – 9x + 20
Create your table.
Product of the
first and last
coefficients
Signs need to
be different
since number
is negative.
Multiply
+20
-1, -20
1, 20
-4, -5
4, 5
-2, -10
2, 10
Add
-9
-21
21
-9
9
-12
12
Middle
coefficient
3) Create your two sets for the factors.
-4 and -5
(x - 4)(x - 5)
x2 – 9x + 20 = (x - 4)(x - 5)
Check your work using FOIL.
Here are some hints to help you
choose your factors in the table.
1) When the last term is positive, the factors
will have the same sign as the middle term.
2) When the last term is negative, the factors
will have different signs.
M
A
4) Factor x2 – 2x – 63
Create your table.
Product of the
first and last
coefficients
Signs need to
be different
since number
is negative.
Multiply
-63
-63, 1
-1, 63
-21, 3
-3, 21
-9, 7
-7, 9
Add
-2
-62
62
-18
18
-2
2
Middle
coefficient
3) Create your two sets for the factors.
-9 and 7
(x - 9)(x + 7)
x2 – 2x – 63 = (x - 9)(x + 7)
Check your work using FOIL.
4)Here’s the grouping method for one with negatives.
Replace the middle term with our factors.
x2 – 2x – 63
x2 – 9x + 7x – 63
Group the terms.
(x2 – 9x) (+ 7x – 63)
Factor out the GCF
x(x – 9) +7(x – 9)
(x + 7)(x – 9)
Factor
2
d
+ 17d – 60
Factor
2
p
– 3p – 40