Teaching Notes 5.19: Factoring the Sums and Differences
of Cubes
Factoring the sums and differences of cubes often presents difficulties for students. Some
students have trouble identifying the cubes of numbers and others confuse the formulas.- Explain that the sums and differences of cubes are binomials and that there are specific for-
mulas for factoring the sums of cubes and the differences of cubes. - Explain that students must be familiar with cubes of numbers in order to apply these for-
mulas. Remind them that the cube of a number is a number raised to the third power. Offer
these examples: 1^3 =1, 2^3 =8, 3^3 =27, 4^3 =64, 5^3 =125, 6^3 =216, 7^3 =343, 8^3 =512,
93 =729, and 10^3 =1, 000. You might find it helpful to expand the notation, for example,
43 = 4 · 4 · 4 =64. - Review the formulas, information, and example on the worksheet with your students. Espe-
cially emphasize that they can check theirwork by multiplying the factors. Depending on
the abilities of your students, you might find 5.9: ‘‘Multiplying Two Polynomials’’ helpful.
EXTRA HELP:
The formulas on the worksheet only apply to the sums or differences of cubes.ANSWER KEY:
(1)(x−2)(x^2 + 2 x+4) (2)(x+5)(x^2 − 5 x+25) (3)Cannot be factored
(4)(x−1)(x^2 +x+1) (5)(4−y)(16+ 4 y+y^2 ) (6)(xy+8)(x^2 y^2 − 8 xy+64)
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(Challenge)Jasmine’s factors were correct. The check was wrong. She should not have used
FOIL to check her work. Therefore, her answer did not check.
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