Factoring Using the Distributive Property

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Write the standard form of an
equation of a line passing through (4,3) with a slope of -2.
Write the equation in standard form
with integer coefficients
y = -1/3x - 4
Do Now:
Worksheet – Match
the correct bottle with
its graph
Factoring Using the
Distributive Property
GCF and Factor by Grouping
1) Factor GCF of 12a2 + 16a


12a2 =2  2  3  a  a
16a =
2 2  2  2  a
2  2  a = 4a
12a  16a 
2
Use
distributive
property
4a (3a) 4a (4)
 4a (3a  4)
PRIME POLYNOMIALS
A POLYNOMIAL IS PRIME IF IT IS NOT THE
PRODUCT OF POLYNOMIALS HAVING INTEGER
COEFFICIENTS.
TO FACTOR A PLYNOMIAL COMPLETLEY,
WRITE IT AS THE PRODUCT OF
•
MONOMIALS
•
PRIME FACTORS WITH AT LEAST TWO
TERMS
TELL WHETHER THE POLYNOMIAL IS
FACTORED COMPLETELY
2X2 + 8 = 2(X2 + 4)
YES, BECAUSE X2 + 4 CANNOT BE FACTORED
USING INTEGER COEFFICIENTS
2X2 – 8 = 2(X2 – 4)
NO, BECAUSE X2 – 4 CAN BE FACTORED AS
(X+2)(X-2)
Using GCF and Grouping to Factor a
Polynomial
2) Factor (4ab  8b)(3a  6)
4b (a  2) 3(a  2)
(4b  3) (a  2)



First, use parentheses to group terms with
common factors.
Next, factor the GCF from each grouping.
Now, Distributive Property…. Group both GCF’s.
and bring down one of the other ( ) since
they’re both the same.
Using GCF and Grouping to Factor a
Polynomial
3) Factor (6 x 15x)(8x  20 )
2
3x (2 x  5) 4 (2 x  5)
(3x  4) (2 x  5)



First, use parentheses to group terms with
common factors.
Next, factor the GCF from each grouping.
Now, Distributive Property…. Group both GCF’s.
and bring down one of the other ( ) since
they’re both the same.
Using GCF and Grouping to Factor a
Polynomial
4) Factor (2a  6a)(3a  9)
2
2a (a  3) 3 (a  3)
(2a  3) (a  3)
First, use parentheses to group terms with
common factors.
Next, factor the GCF from each grouping.
Now, Distributive Property…. Group both GCF’s.
and bring down one of the other ( ) since
they’re both the same.
Using the Additive Inverse Property to
Factor Polynomials.

When factor by grouping, it is often
helpful to be able to recognize
binomials that are additive inverses.


7 – y is
y – 7
 By rewriting 7 – y as -1(y – 7)
8 – x is
x – 8
 By rewriting 8 – x as -1(x – 8)
Factor using the Additive Inverse Property.
5) Factor (35 x  5 xy)(
 3 y  21)
5x (7  y ) 3( y  7)
5x (1) ( y  7) 3 ( y  7)
5x ( y  7) 3 ( y  7)
(5x  3) ( y  7)
Factor using the Additive Inverse Property.
6) Factor( c  2cd)(8d  4 )
c (1  2d ) 4 (2d 1)
c (1)(2d 1) 4 (2d 1)
c(2d 1) 4 (2d 1)
(c  4) (2d  1)
There needs to be a + here
so change the minus to a
+(-15x)
7) Factor 10 x 14 xy 15x  21y
2
(10x 14xy)((15x)  21y)
•Now group your
2
common terms.
•Factor out each
sets GCF.
•Since the first
term is negative,
factor out a
negative number.
•Now, fix your
double sign and
put your answer
together.
2x(5x  7 y) (3)(5x  7 y)
(2 x  3) (5x  7 y)
There needs to be a + here
so change the minus to a
+(-12a)
8) Factor 8ax  6 x  12a  9
8
ax

6
x

(

12
a
)

9
(
)
(
)
•Now group your
common terms.
•Factor out each
sets GCF.
•Since the first
term is negative,
factor out a
negative number.
•Now, fix your
double sign and
put your answer
together.
2x (4a  3) (3) (4a  3)
(2 x  3) (4a  3)
Summary

A polynomial can be factored by
grouping if ALL of the following
situations exist.



There are four or more terms.
Terms with common factors can be
grouped together.
The two common binomial factors
are identical or are additive
inverses of each other.
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