Transcript 3.1 Solving Equations Using Addition and Subtraction

8-1 and 8-2 Factoring Using
the Distributive Property
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
You have used the distributive property to determine a
product – for example:
5xx  2  5x2  10 x
You can also use the distributive property to take the
product and return it to factored form – for example:
5x2  10 x  5xx  2
Today you will use the distributive property to factor out
constants and or variables that are common terms of a
polynomial.
A polynomial is prime if it cannot be factored using integer
coefficients. To factor a polynomial completely, write it
as the product of a monomial and prime factors.
Find the greatest monomial factor. Then factor it out of
the expression.
2x2  8x
2  x  x  2  24 2  x
Write as prime factors.
Circle common primes

2x 
Find the GMF (multiply
the common primes).
Use the distributive property
to factor out the GMF
Whatever is
NOT circled
goes in
parentheses.
Check – Multiply the factors together
using the distributive property.
2xx  4  
 2x2  8x
Find the greatest monomial factor. Then factor it out of
the expression.
2x2  8x
2x x  4
The problem.
Think of the GCF.
Use the distributive property
8x
2x2
to factor out the GCF
2x
2x
You are using
division when you
factor the GCF
out of the
expression!
Find the greatest monomial factor. Then factor it out of
the expression.
3a3  9ab
3  a  a aa2 3  3  a3b
b
Write as prime factors.
Circle common primes

3a 
Find the GMF (multiply
the common primes).
Use the distributive property
to factor out the GMF
Whatever is
NOT circled
goes in
parentheses.
Check – Multiply the factors together
using the distributive property.


3a a2  3b 
 3a3  9ab
Find the greatest monomial factor. Then factor it out of
the expression.
3a3  9ab
The problem.
3a a2 3b
Think of the GMF.
Use the distributive property
9ab
3a3
to factor out the GMF
3a
3a
Find the greatest monomial factor. Then factor it out of
the expression.
25m  21n
5 5  m  7 3 n
Write as prime factors.
Circle common primes
prime
Find the greatest monomial factor. Then factor it out of
the expression.
Example 1
33x3  121x2
Example 2
6x2  3x
Check – Multiply the factors together
using the distributive property.
Example 1 Find the greatest monomial factor. Then
factor it out of the expression.
33x3  121x2
3  11  x  x3x
 x  11 11  x  x
11x 2

33x3  121x2
11x2
(3x  11 )
33x3
2
2
121x
Check – Multiply the factors
together
11x
using the distributive property.
11x 2

2
11x 3x  11   33x3  121x2
Example 2 Find the greatest monomial factor. Then
factor it out of the expression.
6x2  3x
2  3  x2x
 x  31 x
3x 
6x2  3x
3x

(2x  1 )
6x 2
3x
3x together
Check – Multiply the factors
3x
using the distributive property.
3x2x  1 6x2  3x
Find the greatest monomial factor. Then factor
it out of the expression.
Example 3 14x3  21x2
Example 4 5x3  25x2  30x
Example 5 4x3  20x2  24x
Example 6 2n3  4n2  2n
7x2 2x  3


4xx2  5x  6
2nn2  2n  1
5x x2  5x  6
Using the distributive property to factor polynomials
having four or more terms is called factoring by grouping
because pairs of terms are grouped together and
factored. The distributive property is then applied a
second time to factor a common binomial factor.
4ab  8b  3a  6
The problem.
Group terms with
4ab  8b  3a  6
common factors.
4b a  2  3 a  2
Factor the GMF
from each group.

a  2
Factor the common
binomial factor.
Check – Multiply the factors together using FOIL.
Sometimes you can group terms in more than one way
when factoring a polynomial. Here is an alternate way to
group the previous problem.
The problem.
Group terms with
common factors.
Factor the GMF
from each group.
Factor the common
binomial factor.
4ab  8b  3a  6
4ab  3a  8b  6
a 4b  3 2 4b  2
4b  2

Notice that this result is as the previous grouping.
Factor the polynomial.
Example 7
6x2  15x  8x  20
Example 9
2xy  7x  2y  7
Example 8
xy  5y  x  5
Example 10
35x  5xy  3y  21
Check – Multiply the factors together using FOIL.
Factor the polynomial.
Example 8
Example 7
6x2  15x  8x  20
xy  5y  x  5
6x2  15x   8x  20
xy  x  5y  5
3x 2x  5  42x  5
x y  1   5 y  1 

2x  5
2x  53x  4
Check
Undo
double
– Multiply
sign!
y  1

the factors together using FOIL.
Factor the polynomial.
Example 9
2xy  7x  2y  7
Example 10
35x  5xy  3y  21
2xy  2y  7x  7 
35xy  5xy  3y  21
2y x  1  7 x  1
5x 7  y   3 y  7 
x  1

5x  1y  7   3 y  7 
 5xy  7   3y  7 
y  7  5x  3
Check – Multiply the factors together using FOIL.
8-A3 Page 423 # 19–27,
and Page 429 # 9–20.