Transcript Factoring GCF and Grouping

Warm up: p. 479 #72-86 even
Title: Factoring Using the
Distributive Property
EQ: How do we factor polynomials by
using the distributive property? How
do we solve quadratic equations of the
form ax + bx + c?
Factoring by using the distributive
property
• Express a polynomial as the product of a
monomial factor and a polynomial factor.
Example:
Factored form:
Notes cont.
• Factoring involves finding the GCF
Example:
• Now you write each term as the product of
the GCF (divide it out)
Example:
• This will give you the factored form of:
Use the Distributive Property to factor
First, find the CGF of 15x and
.
Factor each number.
.
Circle the common prime factors.
GFC:
Write each term as the product of the GCF and its
remaining factors. Then use the Distributive Property
to factor out the GCF.
Rewrite each term using
the GCF.
Simplify remaining factors.
Distributive Property
Answer: The completely factored form of
is
Use the Distributive Property to factor
.
Factor each number.
Circle the common prime factors.
GFC:
or
Rewrite each term using the GCF.
Distributive Property
Answer: The factored form of
is
Use the Distributive Property to factor each polynomial.
a.
Answer:
b.
6
Grouping to factor
• If a polynomial has four or more terms it helps to
group the polynomial and then factor. This
means you take and split the polynomial into
pairs.
HINTS for grouping:
• There are four or more terms
• Terms with common factors should be grouped
together.
• The two common factors are identical or additive
inverses of each other.
Example:
Factor
Group terms with
common factors.
Factor the GCF
from each grouping.
Answer:
Distributive Property
Factor
Answer:
The additive inverse property
• Recognizing the polynomial as additive inverses
can be VERY helpful when factoring by
grouping. Additive inverses are like (-x-7) (7+x).
You know they are additive inverses bc when
you add them together the sum is 0. Parenthesis
are identical except for signs! You need to pull
out a negative in one of the outside numbers
• Example:
Factor
Group terms with common factors.
Parenthesis are identical except for signs!
You need to pull out a negative in one of the
outside numbers
= -3a (-5 + b) + 4 (b – 5)
What is outside goes in one parenthesis and
what is inside goes onto another.
Answer:
Distributive Property
Factor
Answer:
Zero product property
• If the product of two factors is 0, then at least
one of the two factors is 0.
Solve an equation in factored form
• Set up the two binomials so they are equal to
zero and then solve for the variable.
Example:
Solve
Then check the solutions.
If
Property either
, then according to the Zero Product
or
Original equation
or
Set each factor equal to zero.
Solve each equation.
Answer: The solution set is
Check Substitute 2 and
for x in the original equation.
Solve
Answer: {3, –2}
Then check the solutions.
Solve and equation by factoring
• Write the equation so it is in the form of ab=0
• Then solve for x.
Example:
Solve
Then check the solutions.
Write the equation so that it is of the form
Original equation
Subtract
from each side.
Factor the GCF of 4y and
which is 4y.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
0 and
Check by substituting
for y in the original equation.
Solve
Answer:
Factoring
• You always need to look for a GCF
• If you are grouping you need to make sure
you put like terms together before you put
in ( ) That means that you may need to
reorganize the polynomial
• Always look for difference of perfect
squares