Transcript Factor By Grouping

A1.7.4-5
Factor the following
36𝑥 2 − 121
4𝑥 4 − 64
Now we will learn to factor a quadratic
expression
A quadratic polynomial has the form:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
 Teach
me how to factor
There is a pattern that can help us factor :
Multiply:
(2𝑥 + 3)(𝑥 + 4)
2𝑥 + 3 𝑥 + 2𝑥 + 3 4
2𝑥 2 + 3𝑥 + 8𝑥 + 12
What is the product of the first and last term?
What is the product of the middle two terms?
Multiply
(4x+6)(2x-4)
4𝑥 + 6 2𝑥 + (4𝑥 + 6)(−4)
8𝑥 2 + 12𝑥 − 16𝑥 − 24
What is the product of the first and last terms?
What is the product of the middle terms?
Before combining your like terms, the product
of the middle terms will always equal the
product of your first and last terms
Who wants to see why?
(ax+b)(cx+d)
(𝑎𝑥 + 𝑏)𝑐𝑥 + (𝑎𝑥 + 𝑏)𝑑
𝑎𝑐𝑥 2 + 𝑏𝑐𝑥 + 𝑎𝑑𝑥 + 𝑏𝑑
What is the product of the first and last terms?
𝑎𝑐 ∗ 𝑏𝑑 = 𝑎𝑐𝑏𝑑 = 𝑎𝑏𝑐𝑑
What is the product of the middle terms?
𝑏𝑐 ∗ 𝑎𝑑 = 𝑏𝑐𝑎𝑑 = 𝑎𝑏𝑐𝑑
Multiplication is
Commutative!
2
𝑎𝑥
+ 𝑏𝑥 + 𝑐
Find 2 numbers
that add to this
That have the
same product as
these 2
When we find the right pair, we will split the middle term into
two terms with those coefficients
We are going to learn how to factor a
quadratic to the product of 2 binomials
(2𝑥 + 3)(𝑥 + 4)
2𝑥 + 3 𝑥 + 2𝑥 + 3 4
Today we will start
with this
2𝑥 2 + 3𝑥 + 8𝑥 + 12
2𝑥 2 + 11𝑥 + 12
And turn it into this
2𝑥 2 + 7𝑥 + 5
What is the product of the first and last term?
2 ∙ 5 = 10
What are the factors of 10?
10 and 1, 5 and 2
Do either of these sets have a sum of 7?
5 and 2
Find the two middle terms
8𝑦 2 + 16𝑦 + 6
8 ∗ 6 = 48
Now that you found the two middle terms
You can rewrite the quadratic with the new
terms
8𝑦 2 + 12𝑦 + 4𝑦 + 6
48
factors
16
sum
1, 48
49
2, 24
3, 16
26
19
4, 12
16
 If
you are looking for two numbers that have
a negative product what do you know about
the sign of the two numbers?
Ex: 𝑎 ∗ 𝑏 = −6
a is negative or b is negative (not both)
If their sum is positive, the larger number is positive
If their sum is negative, the larger number is negative
 What
if the product is positive but the sum is
negative?
𝑎 ∗ 𝑏 = 24
𝑎 + 𝑏 = −10
A and b are both negative!
Find the middle two terms:
8𝑥 2 − 18𝑥 + 9
factors
sum
Find the middle two terms:
3𝑝2 − 25𝑝 + 16
factors
sum
Tomorrow we will learn how to factor after we
split the middle term
 A.6.5
Day 1
Find the middle two terms for the following:
2𝑥 2 + 7𝑥 + 5
factors
sum
To factor this we will Factor by Grouping
2𝑥 2 + 5𝑥 + 2𝑥 + 5
Step 1: split the expression into two groups
Step 2: Look at each group separately and factor out the GCF
The GCF might be 1!
2𝑥 2 + 2𝑥 + 5𝑥 + 5
Both terms have an
2𝑥(𝑥
(𝑥 + 1). We can factor
out an (𝑥 + 1)
+ 1) + 5(𝑥 + 1)
(𝑥 + 1)(2𝑥 + 5)
Factor each!
3𝑝2 + 8𝑝 + 5
15
factors
8
sum
7𝑝2 − 23𝑝 − 20
140
factors
-23
sum
7𝑝2 − 24𝑝 + 9
63
factors
-24
sum
 6.5
day 2