Topic 7.1           (Factoring polynomials I)

GREATEST COMMON FACTOR

Find the GCF of:                12x³y and 6xy².

     Find the GCF of   12 and 6, which is    6.

     Is there an x and y in each term?    YES

     Use an x and y in your GCF with the lowest exponent shown in any one term.

     So the GCF of the above problem is      6xy.

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Find the GCF of:         5x³y⁵z²,       15x⁴y⁴z⁴,    and    25x⁵y²z.

     Find the GCF of     5,   15,   and     25, which is 5.

     Is there an x, y, and z in each term? YES

     Use an x, y, and z in your GCF with the lowest exponent shown in any one term.

     So the GCF of the above problem is:        5x³y²z

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Factor:              6xy³ - 4x²y² + 2xy

     The GCF is    2xy.

     Divide 2xy into each term of the problem and put 2xy as a factor in the front.

     Answer:       2xy(3y² - 2xy + 1)

Hint: If there are three terms in your problem, there must be three terms in the parenthesis of the answer.   (2xy/2xy  is 1 not 0) 

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FACTORING BY GROUPING

These problems will have a common factor in the form of a binomial

Sample:            x(y - 2) + 2(y - 2)

     (y - 2) is the common factor of the two terms.

     Factor this out by dividing each term of the problem by (y - 2).

     When you factor (y - 2) out, what’s left is (x + 2).   

     Answer:            (y  - 2)(x  +  2)

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Sample:            a³ - a²b + ab² - b³

       You must group the problem into two groups, usually, but not always, the first

       two and the last two.

                            (a³ - a²b)   +   (ab² - b³)

      Factor the GCF of each group.

      a²(a - b) + b²(a - b)    Do you have two terms with the same binomial factor?

                                                  YES

      Factor that binomial factor out.

      Answer:          (a - b)(a² + b²)