Transcript Document

Unit 6 Factoring Polynomials
Greatest Common Factor
Factoring by Grouping (Four Terms)
AC Method (3 Terms)
Difference of Two Squares (2 Terms)
Sum and Difference of Two Cubes (2
Terms)
Why do I need to be able to factor?
• We need to be able to factor in order
to
Solve quadratic equations.
Work with Rational Expressions
What is factoring?
• Factoring is the OPPOSITE of distribution
• When we factor, we are taking apart a
polynomial
What are factors?
Each piece that we take apart is called a
factor
FACTORS are terms that are multiplied
together to reach a PRODUCT (this is a
code word for multiplication). Factors can
be numbers, variables, and quantities
(binomials or larger within parentheses).
Does every polynomial factor?
• No … some polynomials (or numbers for
that matter) do not factor!
• Things that do not factor are called PRIME
The Factoring Process
• There are MANY ways to teach/discuss
factoring polynomials.
• If you have a way that works for you that
you understand then you should feel no
pressure to do it any other way.
• If you don’t have a method or you are
interested in seeing a different process, I
promise you mine works every time.
CARA’S PROCESS FOR
FACTORING POLYNOMIALS
Factor out the GCF first
There might not be a GCF (other than 1).
Just because there is a GCF factored out does
not mean you are done.
Just because you did not factor out a GCF does
not mean the polynomial is prime.
GCF can be a number, a letter, a number-letter
combination, or a polynomial.
WARNING
Factoring out the GCF does not mean you
are done factoring … it is only step 1!
You might be done … but you might not!
Just in case I was not clear earlier!
CARA’S PROCESS FOR
FACTORING POLYNOMIALS
Count the number of terms
If 4 terms: Use the grouping method
If 3 terms: Use the AC method
If 2 terms: Use the recipe method
CARA’S PROCESS FOR
FACTORING POLYNOMIALS
Last step … double check to make sure
none of the polynomial terms can be
factored (this is especially important if
you did not factor a GCF out first).
As a reminder … you should be able to
multiply your factors and get a product
that is equal to the original polynomial.
Example of GCF of Polynomial
Factor 7x2 - 14xy
1. Find the GCF of the terms 7x2 and
14xy
7x2 =
14xy =
The GCF is:
Example of GCF of Polynomial
Factor 7x2 - 14xy
1. Find the GCF of the terms 7x2 and 14xy
The GCF is 7x
2. Divide each term in the polynomial
by the GCF you just chose.
7x2
-14xy
Answer:
Example of GCF of Polynomial
Factor 7x2 - 14xy
1. Find the GCF of the terms 7x2 and 14xy
The GCF is 7x
2. Divide each term in the polynomial by the GCF you just
chose. Answer: (x – 2y)
3. Write the GCF followed by the result
from step 2 IN PARENTHESES.
Try This One:
12a2 + 16ab – 12a2b
Four Terms






Factor out GCF FIRST
Split the polynomial into 2
groups
Factor out the GCF in the
left group
Factor out the GCF in the
right group
Look at the two groups and
factor out the common factor
from both
Write the remainder as the
2nd factor
4 Term Polynomials
Example: Factor ax + ay + 5x + 5y
4 Term Polynomials
Example: Factor ax + ay + 5x + 5y
1.
Factor out GCF: There was none.
2.
Make two groups
3.
Factor the GCF from each pair.
4.
Factor the common factor from the two groups.
a(x + y) + 5(x + y)
a(x + y) + 5(x + y)
(x + y)(a + 5) OR (a + 5)(x + y)
4 Term Polynomials
Example: Factor 5a - 2x + 10 - ax
4 Term Polynomials
Example: Factor 5a - 2x + 10 - ax
1.
Factor out GCF: There was none.
2.
Make two groups
3.
Factor the GCF from each pair.
4.
Factor the common factor from the two groups.
a(5 - x) + 2(5 - x)
a(5 - x) + 2(5 - x)
(a + 2)(5 - x) OR (5 - x)(a + 2)
As we transition from FOUR term
polynomials to THREE term
polynomials … we are going to use our
new GROUPING method.
3 Term Polynomials
AC METHOD (FACTORING 3 TERM POLYNOMIALS)
 STEP1: Make sure the polynomial is in standard form (exponents in
descending order) …
 STEP2: Factor out the GCF
 STEP3: Label the value for a, b, and c
 STEP4: We are trying to find a pair of numbers that will meet two specific
conditions:
In order to do this, I am going to write out as many pairs of numbers I
can that will give me the product ac and then from this list, find the pair of
numbers that will give me a sum equal to b.
 STEP5: Rewrite the middle term of the trinomial using the two terms you
found so that you have a 4 Term Polynomial
 STEP6: Factor this four term polynomial using the GROUPING technique
 PULL THIS PAGE FROM YOUR NOTES SO YOU CAN
USE IT AS A GUIDE!
Let’s work this One:
2x2 + 5x + 3
Try This One:
3x2 - x - 14
2 Terms
• After you factor out the GCF, polynomials
with 2 terms factor one of three ways
– Difference of squares
– Difference of cubes
– Sum of cubes
Difference of Squares
Meaning: A perfect square minus a perfect
square
Form: F2 – L2
Factors as: (F + L)(F – L)
Examples: x2 – 9 = x2 – 32=(x + 3)(x – 3)
y2 – 36 = y2 – 62=(y + 6)(y – 6)
Try This One:
81y2 - 4
Sum and Difference of Cubes
Song: “A little one, a little bigger one, first
one, second one, first one squared, product
of the two and the second one square. The
first sign is the same, second is the opposite
and the last one is always plus.”
Difference of Cubes
Meaning: A perfect cube minus a perfect
cube
Form: F3 – L3
Factors as: (F - L)(F2 + FL + L2)
Examples:
x3 – 8 =
x3 – 64 =
Sum of Cubes
Meaning: A perfect cube plus a perfect cube
Form: F3 + L3
Factors as: (F + L)(F2 - FL + L2)
Examples:
x3 + 8 =
x3 + 64 =
Sum of Squares
Meaning: A perfect square plus a perfect
square
Form: F2 + L2
Factors as: DOES NOT FACTOR … do not
be tempted!
Examples: x2 + 4
x2 + 100
The KEY to these are knowing
what perfect squares and cubes
are
PERFECT SQUARES
1
4
9
16
25
36
49
64
81
100 etc
PERFECT CUBES
1
8
27
64
125 etc
30x3 - 25x2y - 30xy2
8y2 + 10y - 12