3.1 Solving Equations Using Addition and Subtraction
Download
Report
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:
5xx 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 5xx 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.
2xx 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.
3x2x 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
4xx2 5x 6
2nn2 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 42x 5
x y 1 5 y 1
2x 5
2x 53x 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 1y 7 3 y 7
5xy 7 3y 7
y 7 5x 3
Check – Multiply the factors together using FOIL.
8-A3 Page 423 # 19–27,
and Page 429 # 9–20.