Teaching Notes 5.13: Factoring Polynomials by Finding
the Greatest Monomial Factor
Factoring a polynomial by using the greatest monomial factor is difficult for many students. They
may fail to find the greatest monomial factor or they may make a mistake in division.- Explain that although the term ‘‘greatest common factor’’ applies to numbers and monomials,
the greatest monomial factor applies to polynomials. It is the greatest common monomial
factor of each of the terms of the polynomial. - Provide examples of the greatest monomial factor such as the following:
- 4 x^2 + 7 x;xis the greatest monomial factor.
- 12 a^2 + 16 ab; 4 ais the greatest monomial factor.
- 3 x^2 + 2 b;1 is the greatest monomial factor.
- Explain that because the greatest monomial factor of a polynomial is a factor of each term,
the other factors can be found by dividing the polynomial by the greatest monomial fac-
tor. Using the first example above, divide each term of the polynomial byx.4 x^2
x+
7 x
x=
4 x+7 Thus, the factors of 4x^2 + 7 xarexand (4x+7). Using the same procedure in the sec-
ond example, the factors of 12a^2 + 16 abare 4aand (3a+ 4 b). Explain that 3x^2 + 2 bcannot
be factored because 1 is the greatest monomial factor.- Review the information and example on the worksheet with your students. If necessary,
discuss in detail and demonstrate the steps for finding the greatest monomial factor and
factoring the example.
EXTRA HELP:
Be sure you have found the greatest monomial factor before you divide.ANSWER KEY:
(1)3(x+5) (2)Cannot be factored (3)ab(8− 3 b) (4) 10 x(2xy− 1 + 4 x^2 )
(5)x(49x−1) (6) 5 x^2 y^3 (7x^2 − 3 y^2 ) (7)a^2 b(1+ 2 ab) (8) 6 x(2y−x)
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(Challenge)Eva is incorrect. 2xyis not the greatest monomial factor; 2xy^2 is. The correct
factorization is 2xy^2 (2x−5).
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