CHAPTER 7: FACTORING POLYNOMIALS

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1 CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor an epression is to rewrite it as a product of or more quantities. Factoring is sometimes called FACTORIZATION. 1 can be FACTORED into the product 3*. Variable Epressions can also be factored. 3 can be factored into 7( ) or (7) or (7 ), etc How about this polnomial epression? To factor a polnomial, the FIRST STEP is to look for a GREATEST COMMON FACTOR. The GCF of a polnomial is the GREATEST number (or variable epression) that is a factor of ever term in the epression. That is, it is the variable epression that is the GCF of the coefficients, and the GCF of each of the variables. How man terms are there? What are the? What is the GCF of these terms? Let s take another look at the polnomial: Coefficients: The Coefficients are, -3, and 10. The Coefficient GCF is. Variables: The onl variable is, the first term has 3, the nd term has, the third term has. The GCF is the greatest power that can go into ALL of those terms, but practicall this means it will be the variable term with the smallest power: The GCF of is Now to factor the polnomial: Rewrite the polnomial as a product with as one factor, and the remaining epression (after dividing each term b ) This polnomial cannot be factored an more. It is prime. We ll see wh later.

2 GCF :,, : ' GCF :,, : ' GCF : 1 Coefficients :16,8, and Eample: s s Put it all together and the GCF of the polnomial is: NOW YOU TRY: Factor this:

3 Tr factoring 3(-) (-) FACTORING BY GROUPING Eample: ( + ) + 3( + ) Remember, a FACTOR is something being multiplied in a product. Do ou see a common factor in this epression? ( + ) + 3( + ) COMMON FACTOR ( ) ( ) ( )( 3) 3( ) Eample : ( - ) + ( - ) At first, it looks like there is no common factor, but notice that - and - are ver similar. In fact, - (-) = -+ = - So we can rewrite (-) has (-) ( - ) + (-( - )) = ( - ) ( - ) =( )( )

4 Eample : If there is not a common factor of ALL the terms, ou can factor b GROUPING the terms in to groups that DO have a common factor. (3 3 ) + ( 6 + 8) = (3 - ) + -(3 ) Factor out a - instead of so that this group will have a common factor to the other group (3 ). = (3 - )( ) YOU TRY FACTORING

5 FACTORING POLYNOMIALS OF THE FORM + b + c Eample 1: Factor the polnomial: STEP 1: Is there a GCF of all three terms? NO STEP : Is this polnomial a trinomial with degree? YES Since this is a trinomial with degree, it is possible that this polnomial can be factored into binomials: ( + a )( + b ) Remember, the FOIL method for ( + a)( + b) = + a + b + ab = + (a + b) + ab So from this general form, we see that the factors of the Last Term (ab), must add up to make the Middle Term s coefficient. STEP 3: Tr different factor s of the Last Term that will add up to the Middle Term s coefficient. Polnomial: Last Term: Middle Term s Coefficient: Factors of Last Term Sum of Those Factors 1, = 33,8 +8 = 1, = 18 FACTORIZATION: ( + )( + 16)

6 Eample : Factor 6 16 STEP 1: Is there a GCF of all three terms? STEP : Is it a trinomial with degree? Last Term: Middle Term s Coefficient: Factors of Last Term Sum of Those Factors 1, = -1-1, = 1, = -6 -, = 6, = 0 -, - + = 0 FACTORIZATION: ( )( +8) UNFACTORABLE TRINOMIALS: 6 8 There are no factors of -8 that can add up to -6, So this is considered a prime polnomial and is nonfactorable over the integers.

7 Eample: Factor 3a b 18ab 81b STEP 1: Is there a GCF of all three terms? YES GCF is Factor out the GCF: STEP : After factoring out the GCF is one of the factors a trinomial with degree? STEP 3: Find factors of the last term of the trinomial that add up to the middle term s coefficient and factor into two binomials. STEP : Don t forget STEP 1 S GCF in ou final factorization!

8 EXAMPLE : Factor STEP 1: Is there a GCF of all terms? NO STEP : Is this a trinomial with degree? YES STEP 3: Find factors of the last term of the trinomial that add up to the middle term s coefficient and factor into two binomials. In this case, make sure the factors of the the last term are like terms that can be combined. (eg. 0 and 1 are not like terms but are factors of 0 ) Factors of Last Term, 0 Sum of Those Factors 0, =1, = 1, + = 9 FACTORIZATION: ( + )( + )