Teaching Notes 5.4: Subtracting Polynomials
To subtract polynomials, students must master the concept of rewriting subtraction as addition
and changing the sign of each of the terms in the polynomial they are subtracting. A common
problem is forgetting to change the signs of the terms.- Provide this example: A teacher is distributing worksheets and calculators at the beginning
of class. She has 25 worksheets and 25 calculators. She hands out 23 worksheets and provides
8 calculators for students who need them. Ifwrepresents the number of worksheets and
crepresents the number of calculators, (25w+ 25 c)−(23w+ 8 c) represents the number of
items that are left over. Explain that to findthe difference students must rewrite the problem
as (25w+ 25 c)+(− 23 w+(− 8 c)) and then add the polynomials. 25w+(− 23 w)+ 25 c+
(− 8 c)= 2 w+ 17 c. Thus, the teacher has 2 worksheets and 17 calculators that she did not
distribute. - Review the information and examples on the worksheet with your students. Emphasize that
they must rewrite the problem by adding the opposite of each term of the polynomial they
are subtracting. Make sure that your students recognize the steps involved in rewriting.
After rewriting the polynomial to be subtracted, they must combine similar terms by adding
the coefficients as they would add integers. Also note that constants must be added, follow-
ing the rules for adding integers. If necessary, review the steps for adding and subtracting
integers.
EXTRA HELP:
The opposite of a positive coefficient is a negative coefficient and the opposite of a negative
coefficient is a positive coefficient.ANSWER KEY:
(1) 5 x+ 5 (2)− 7 x^2 + 11 (3)− 9 y^2 + 2 (4) 2 x^2 + 4 xy−y+ 8
(5)−x^2 + 9 x− 11 y^2 (6)− 4 a^2 − 11 ab+ 16 (7)− 2 y^2 − 10 y− 6 (8) 2 a+ 7 b− 2 c+ 1
(9) 24 a^2 − 7 ab+ 5 (10)− 2 y^2 + 6 y+ 13
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(Challenge)Explanations may vary. One possible answer follows. Yes, if the coefficients of the
similar terms are the same, the difference will be a constant, for example (x−4)−(x−8)=4.
In this case, the answer is a constant.
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