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Module 7: Algebra

Dividing Polynomials and Factoring

These next set of skills are, more or less, the opposite of the previous section in Expanding Polynomials and Distribution. Much like how division is the opposite of multiplication, these skills are essentially opposite operations.

The opposite of distribution is factoring, the act of pulling a common factor (GCF) from a polynomial to create the multiplication statement we saw in the last section. Note that that is how you write a factored polynomial, as the factor pulled out does not get eliminated. Furthermore, remember that both constants and variables can be factored out.

Tutorial 29: How to Factor a Term Out of Polynomials

Example:

( 5 x 2 + 15 x y − 40 x ) {"version":"1.1","math":"(5x^2+15xy-40x)"}

  1. Determine the GCF of each term in the polynomial (constants and variables).

    For example: The GCF of 5 , 15 , {"version":"1.1","math":"5, 15, "} and − 40 {"version":"1.1","math":"-40"} is 5 {"version":"1.1","math":"5"}. For the variables, x {"version":"1.1","math":"x"} appears in every term at least once, making it also a factor of each term. Thus, the GCF of the polynomial is 5 x {"version":"1.1","math":"5x"}

  2. Divide each term in the polynomial by the GCF to pull it out of the polynomial. Place brackets around the simplified polynomial and place the GCF on the outside.

    For example:

    5 x 2 + 15 x y − 40 x {"version":"1.1","math":"5x^2+15xy-40x"}

    → 5 x 2 5 x + 15 x y 5 x − 40 x 5 x = x + 3 y − 8 {"version":"1.1","math":"\rightarrow \dfrac{5x^2}{5x} + \dfrac{15xy}{5x} - \dfrac{40x}{5x} = x+3y-8 "}

    ∴ 5 x ( x + 3 y − 8 ) {"version":"1.1","math":"\therefore 5x(x+3y-8)"}

This coincides the act of reducing polynomials in fractions. Let’s say we wanted to reduce the following fraction, how is it done?

105 x 2 + 35 x − 15 x y 20 x y {"version":"1.1","math":"\dfrac{105x^2+35x-15xy}{20xy} "}

To solve simplify this, we can split the fraction in to one fraction per term of the trinomial, reduce each by the denominator, then form the common denominator:

= 105 x 2 20 x y + 35 x 20 x y − 15 x y 20 x y {"version":"1.1","math":"= \dfrac{105x^2}{20xy} + \dfrac{35x}{20xy} - \dfrac{15xy}{20xy} "}

= 21 x 4 y + 7 4 y − 3 4 {"version":"1.1","math":"= \dfrac{21x}{4y} + \dfrac{7}{4y} - \dfrac{3}{4} "}

= 21 x 4 y + 7 4 y − 3 y 4 y {"version":"1.1","math":"= \dfrac{21x}{4y} + \dfrac{7}{4y} - \dfrac{3y}{4y} "}

= 21 x + 7 − 3 y 4 y {"version":"1.1","math":"= \dfrac{21x + 7 - 3y}{4y} "}

Or, we could factor a term out of the trinomial and reduce where possible:

= 5 x ( 21 x + 7 − 3 y ) 20 x y {"version":"1.1","math":"= \dfrac{5x(21x+7-3y)}{20xy} "}

= 5 x ( 21 x + 7 − 3 y ) 20 x y {"version":"1.1","math":"= \dfrac{5x(21x+7-3y)}{20xy} "}

= ( 21 x + 7 − 3 y ) 4 y {"version":"1.1","math":"= \dfrac{(21x+7-3y)}{4y} "}

Exponent rules and reduction rules apply to both terms and whole polynomials.

For example, in ( y − 4 ) 2 ( y + 2 ) ( y − 4 ) , ( y − 4 ) {"version":"1.1","math":"\dfrac{(y-4)^2(y+2)}{(y-4)},\;(y-4) "} is reducible to produce ( y − 4 ) ( y − 2 ) {"version":"1.1","math":"(y - 4) (y-2)"}. This is comparable to 25 ( 2 ) 5 {"version":"1.1","math":"\dfrac{25(2)}{5} "} which gives 5 ( 2 ) → 10 {"version":"1.1","math":"5(2)\rightarrow 10"} after reduction.

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