Yours
Truly answers: Just like with squares, we have both expansion & factorization formulas. Usually, it is easier to expand an expression than factoring one because, in the worst case, we can multiply an expression by itself three times.
| (1) |
Expansion Formulas |
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(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3
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In the second formula, notice where the minus is. At least once in our career, we should multiply
a – b by itself three times by hand and convince ourselves that the second formula is indeed correct.
Another thing to remember is how the coefficients line up on the right side of each formula.
If we disregard ±, then the coefficients are
1 3 3 1
Since they line up symmetrically, it is easy to remember them. |
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| (2) |
Factorization Formulas |
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a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
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These formulas do require memorization. But, we should expand the right-hand side and convince ourselves that they are indeed correct formulas. |
| Example |
Factor 8x3 – 27. |
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(Solution)
We first note that both 8 and 27 are cubes. So, 8x3 – 27 = (2x)3 – 33. Now, use the last formula with a = 2x and b = 3. Thus,
8x3 – 27 = (2x)3 – 33 = (2x – 3)[(2x)2 + (2x)(3) +32)] = (2x – 3)(4x2 + 6x + 9).
Done.
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