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.

NIKITA’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!

NIKITA’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

NIKITA’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 7x xy 1.Find the GCF of the terms 7x2 and 14xy 7x 2 = 14xy = The GCF is:

Example of GCF of Polynomial Factor 7x xy 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. 7x 2 -14xy Answer:

Example of GCF of Polynomial Factor 7x xy 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.

4 Term Polynomials Example: Factor ax + ay + 5x + 5y

4 Term Polynomials Example: Factor ax + ay + 5x + 5y If you look carefully, there is no GCF larger than 1 common to all FOUR terms. Since we never factor out a plain 1 (since it changes nothing), we need to find another way to approach this problem.

4 Term Polynomials Example: Factor ax + ay + 5x + 5y 1.Factor out GCF: There was none. 2.Pair up the terms in such a way that each pair has something in common. Many times the pairs work out to be the first two and the second two, but don’t be afraid to use the commutative property and shuffle things around! Then make TWO groups by drawing a vertical line separating the first two terms with the second two terms. 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, as if they were completely different problems. 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. a(x + y) + 5(x + y) 4.Factor the common factor from the two groups. a(x + y) + 5(x + y) (x + y)(a + 5) OR (a + 5)(x + y)

4 Term Polynomials Example: Factor 5a - 2x ax

4 Term Polynomials Example: Factor 5a - 2x ax If you look carefully, there is no GCF larger than 1 common to all FOUR terms. Since we never factor out a plain 1 (since it changes nothing), we need to find another way to approach this problem.

4 Term Polynomials Example: Factor 5a - 2x ax 1.Factor out GCF: There was none. 2.Pair up the terms in such a way that each pair has something in common. Many times the pairs work out to be the first two and the second two, but don’t be afraid to use the commutative property and shuffle things around! Then make TWO groups by drawing a vertical line separating the first two terms with the second two terms. 5a - 2x ax Looking at both the left group and the right group, I do not see any GCF in either group that can be factored out … lets rearrange the terms to see if we can find something that will work. 5a - ax x

4 Term Polynomials Example: Factor 5a - 2x ax 1.Factor out GCF: There was none. 2.Make two groups 3.Factor the GCF from each pair, as if they were completely different problems. 5a - 2x ax

4 Term Polynomials Example: Factor 5a - 2x ax 1.Factor out GCF: There was none. 2.Make two groups 3.Factor the GCF from each pair. a(5 - x) + 2(5 - x) 4.Factor the common factor from the two groups. 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!

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: F 2 – L 2 Factors as: (F + L)(F – L) Examples: x 2 – 9 = x 2 – 3 2 =(x + 3)(x – 3) y 2 – 36 = y 2 – 6 2 =(y + 6)(y – 6)

Difference of Cubes Meaning: A perfect cube minus a perfect cube Form: F 3 – L 3 Factors as: (F - L)(F 2 + FL + L 2 ) Examples: x 3 – 8 = x 3 – 64 =

Sum of Cubes Meaning: A perfect cube plus a perfect cube Form: F 3 + L 3 Factors as: (F + L)(F 2 - FL + L 2 ) Examples: x = x =

Sum of Squares Meaning: A perfect square plus a perfect square Form: F 2 + L 2 Factors as: DOES NOT FACTOR … do not be tempted! Examples: x x

The KEY to these are knowing what perfect squares and cubes are PERFECT SQUARESPERFECT CUBES etc etc

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 2 nd factor

Four Terms  Factor out GCF FIRST In this example there is not one!

Four Terms  Factor out GCF FIRST  Split the polynomial into 2 groups  Yes … it really is that simple!

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

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

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 2 nd factor

Four Terms  Factor out GCF FIRST In this example there is not one!

Four Terms  Factor out GCF FIRST  Split the polynomial into 2 groups  Yes … it really is that simple!

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

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

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 2 nd factor