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Question
If a − 2b + 3c = 0; state the value of a3 − 8b3 + 27c3.
Sum
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Solution
a − 2b + 3c = 0
a3 − 8b3 + 27c3
x3 + y3 + z3 − 3xyz = (x + y + z) (x2 + y2 + z2 − xy − yz − zx)
a3 − (2b)3 + (3c)3 = a3 − 8b3 + 27c3
a3 + (−2b)3 + (3c)3
x + y + z = 0 ⇒ x3 + y3 + z3 = 3xyz
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x = a
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y = −2b
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z = 3c
a − 2b + 3c = 0 ⇒ x + y + z = 0
x3 + y3 + z3 = 3xyz ⇒ a3 + (−2b)3 + (3c)3 = 3(a) (−2b) (3c)
= 3⋅a⋅(−2b)⋅3c = −18abc
a3 − 8b3 + 27c3 = −18abc
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Special Product
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