Transcript Section 5.1 The Greatest Common Factor and Factoring by Grouping

Section
The Greatest Common Factor
and
Factoring by Grouping
Factors
• The answer to a
multiplication problem is
called a product.
• Numbers that will multiply
together to give a product
are called factors of the
product.
• List all of the factors of 36.
1  36 = 36
2  18 = 36
3  12 = 36
4  9 = 36
6  6 = 36
1, 2, 3, 4, 6, 9, 12, 18, 36
List the Factors of 30
1  30 = 30
2  15 = 30
3  10 = 30
5  6 = 30
1, 2, 3, 5, 6, 10, 15, 30
Compare the factors of 30 and
36.
Factors of 30
1, 2, 3, 5, 6, 10, 15, 30
Factors of 36
1, 2, 3, 4, 6, 9, 12, 18, 36
What are
the
common
factors of
30 and 36?
1, 2, 3, 6
6
Which is the
greatest?
6 is the greatest
common factor of
30 and 36
Common Factor
• An integer that is a factor of two or more
integers is a common factor of those
integers.
• 6 is a common factor of 30 and 36.
• 1, 2, and 3 are also common factors of 30
and 36.
• 6 is the greatest common factor of 30 and
36.
Finding the Greatest Common
Factor for Numbers
• Write each number in
factored form.
Example:
• Find GCF of 36 and 48
36 = 12 · 3
48 = 12 · 4
GCF = 12
Find the GCF of 30, 20, 15
30 = 6 · 5
20 = 4 · 5
15 = 3 · 5
Since 5 is the only
common factor it is
also the greatest
common factor GCF.
Find the GCF of 10, 21, 15
10 = 1 · 2 · 5 Since 1 is the only
common factor it is
21 = 1 · 3 · 7
also the greatest
15 = 1 · 3 · 5 common factor GCF.
Usually 1 is not included in the list of
factors but it was needed this time to
show a common factor.
Find the GCF of 6m4, 9m2, 12m5
6m4 = 2 · 3 · m2 · m2
9m2 = 3 · 3 · m2
12m5 = 4 · 3 · m2 · m3
GCF =
2
3m
Find the GCF of 5x and 15
5x = 5 · x
GCF = 5
15 = 3 · 5
5x + 15 = 5( + )
Factored
Form
Distribution is
the way to
check
factoring.
= 5(x + 3) = 5x + 15
Factor out the greatest common
factor.
First write
down the GCF
10y5
-
8y4
+
6y2
=
Divide each
coefficient by 2
2
3
2y (5y
-
2
4y
+ 3)
Keep the
same signs
Factor out the greatest common
factor.
2y2(5y3 - 4y2 + 3)= 10y5 - 8y4 + 6y2
Check with
distribution
Factor out the greatest common
factor.
5
2
8p q
+
6
3
16p q
-
4
7
12p q
=
4p4q2 ( 2p + 4p2q-3q5)
Factor Out the Common Factor
ax + bx = x (a + b)
a(x + 5) + b(x + 5) = (x + 5) (a + b)
Factor Out the Common Factor
x(3a + 2) + 7(3a + 2) = (3a + 2)(x +7)
Factor By Grouping
1. Group First Two Terms and Group
the last Two Terms.
2. Find Greatest Common Factor
of First Two Terms
3. Find Greatest Common Factor
of Second Two Terms
4. Write Down What Is Common to Both
5. Write Down What Is Left Over
Factor by Grouping
2
6y
2
20w
- 8yw + 15yw =
2
2
(6y - 8yw) + (15yw - 20w ) =
2y(3y - 4w) + 5w(3y - 4w) =
(3y - 4w)(2y + 5w) =
Factor 6x2 – 3x – 4x + 2
by grouping
– 3x – 4x + 2
2
= (6x – 3x) + (– 4x + 2)
= 3x(2x – 1) + -2(2x - 1)
= (2x – 1)(3x – 2)
2
6x
Your Turn
Factor xy + 2x + 4y + 8 by grouping
xy + 2x + 4y + 8
= (xy + 2x) + (4y + 8)
= x(y + 2) + 4(y + 2)
= (y + 2)(x + 4)
Try Again
Factor 6x2 – 9x – 4x + 6 by grouping
– 9x – 4x + 6
2
= (6x – 9x) + (– 4x + 6)
= 3x(2x – 3) + -2(2x – 3)
= (2x – 3)(3x – 2)
2
6x
What Did You Learn?
• After factoring a polynomial how
can you check your result?
• When would you try to factor by
grouping?