• Factoring Polynomials

     


    Common Term

    1.  Is there a number that is a factor of all coefficients?  If so, write it down.

    2.  Is there a variable that appears in all terms?  If so, write it down and affix the lowest

         exponent of all exponents for that variable.

    3.  Now open a set of parenthesis next to this common term and divide all the terms of the

         polynomial by the common term and put the resulting terms in the parenthesis.

     

           4x2 – 2x   =    2x(2x – 1)

     

    Grouping

    1.  Group the terms in pairs and factor out common term.

    2.  Do you now have a common binomial term?

    3.  If so, write the common binomial term as the second binomial and the coefficients of the common binomial terms as your first polynomial.

     

           x2 + 2x – 6x – 12          

           x(x + 2) – 6(x + 2)            

           (x – 6)(x + 2)

     

    x2 + bx + c                                          

    1.  Set-up two sets of parenthesis and place “x” in the first slot of each set

    2.  Ask the question, what are the factors of c that combine to equal b?

    3.  Place the factors in the second slot of the two binomials.

     

         x2 + 5x + 4     =   (x + 1)(x + 4)    - The factors of 4 that combine to equal 5  --- +4 and +1

                                                   

         x2 + 5x – 6     =   (x + 6)(x – 1)  - The factors of -6 that combine to equal +5 ---- +6 and -1

     

    ax2 + bx + c                                        

    1.  Multiply “a” times”c” and list the factors of “ac”

    2.  Ask the question, what are the factors of “ac” that combine to equal “b”?

    3.  Re-write the polynomial using the factors from #2 as the coefficients of the term with “b”.

    4.  Factor by grouping

     

         2x2 + 11x + 14             First take 2 times 14 = 28;  What are the factors of "28" that combine to equal "11?"

        2x2 + 4x + 7x + 14          Re-write 11x as "4x + 7x"

         2x(x + 2) + 7(x + 2)       Factor by grouping  

         (2x + 7)(x + 2)

     

    Difference of Squares                                   

    1.  Is the first term a perfect square?

    2.  Is the second term a perfect square?

    3.  Is there a subtraction sign between the two terms?

    4.  If yes to all three, open two sets of parenthesis, put the square root of the first term in

          the first slot of each binomial, the square root of the second term in the second slot,

          and in one binomial put a + sign between the two terms and in the other binomial a

          sign between the two terms.

     

         x2 – 25   =   (x + 5)(x – 5)

    Square of Binomial  --  Perfect Square Trinomial                         

    1.  Is the first term a perfect square?  - must be positive

    2.  Is the last term a perfect square? – must be positive

    3.  If you multiply the square root of the first term times the square root of the last term

          and then by 2, do you get the middle term (disregarding the sign)?

    4.  If yes to all three, open a set of parenthesis, put the square root of the first term in

         the first slot, the square root of the last term in the second slot, and the exponent

         2 outside the parenthesis.

    5.  Then put the sign of the second term of the original trinomial as the sign between the two

         terms of the binomial.

     

         x2 – 12x + 36   = (x – 6)2

     

         x2 + 18x + 81  =  (x + 9)2

     

    Difference of Cubes

    1.  Is the first term a perfect cube?

    2.  Is the second term a perfect cube?

    3.  Is there a subtraction sign between the two terms?

    4.  If yes to all three, open two sets of parenthesis, put the cube root of the first term in the

          first slot of the first binomial the cube root of the second term in the second slot and a

          subtraction sign between the two terms.

    5.  In the second set of parenthesis put the square of the first term of the first binomial, a plus

          sign, then the product of the two terms in the first binomial, another plus sign and the

          square of the second term of the first binomial.

     

         x3 – 125    

         (x – 5)(x2 + 5x + 25)

     

    Sum of Cubes

    1. Same as Difference of Cubes but make a plus sign in the first binomial and a

         subtraction sign for the second term of the second polynomial.

       

         x3 + 27    

         (x + 3)(x2 – 3x + 9)

     

     

     

     

     

     

Last Modified on February 24, 2015