Teaching Notes 5.17: Factoring by Grouping
Factoring by grouping involves finding and factoring the greatest monomial factor of two
binomials and writing the expression as the product of binomials. An error in any part of this
process will result in an incorrect factorization.- Explain that factoring by grouping may be used to factor polynomials that have four terms,
provided that each group of two terms has a common monomial factor. Depending on the
abilities of your students, you may wish to review 5.13: ‘‘Factoring Polynomials by Finding
the Greatest Monomial Factor.’’ - Explain that factoring by grouping requires students to rearrange the terms into two groups
and then find the greatest monomial factor of each group. A binomial that is a common factor
of both groups will result. Note that if a common binomial factor does not result, students
may have to regroup the terms or the polynomial cannot be factored by grouping. - Review the information and example on the worksheet with your students. Note that the
example is factored in two different ways with the same result. Also note that each group
of two terms has a common monomial factor. Students cannot rewrite the example as
(xy+10)+(2x+ 5 y) because neitherxy+10 nor 2x+ 5 yhas a common factor. Explain that
if the binomial is a common factor of the expression, it should be factored out. Remind your
students that when checking the answer in the example,xyis the same asyx, and similar
terms can be added in any order.
EXTRA HELP:
A polynomial can be factored only by grouping if it has a common binomial factor.ANSWER KEY:
(1)(x−2)(y+3) (2)(y+2)(x−3) (3)(y−1)(3x+2) (4)(x+3)(y−7) (5)(x+3)(y−4)
(6)(x^2 +2)(y+2) (7)(x− 2 y)(x−4) (8)(xy−3)(x−4) (9)(2x−1)(y−3)
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(Challenge)The teacher is correct. Had Miguel regrouped the terms as (2x^2 −x)−(20x+10)
and wrotex(2x−1)−10(2x−1), he would have found the correct factorization, which is
(x−10)(2x−1).
------------------------------------------------------------------------------------------208 THE ALGEBRA TEACHER’S GUIDE