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Write in completely factored form: (2x-y)^6 -64x^6?
This is a question on my problem set, I'm supposed to factor it completely.
My answer is (4x-y)(4x^2 -2xy+y^2)(-y(12x^2 +2xy +y^2) but I must have made a mistake somewhere because they don't match up. I began by taking the difference of two perfect squares, and then having to multiply the difference of two perfect cubes by the sum of two perfect cubes. It is very possible there is a mistake in my work, please help
1 Answer
- Anonymous9 years agoFavorite Answer
Since (2x - y)^6 = [(2x - y)^3]^2 and 64x^6 = (8x^3)^2, we can write:
(2x - y)^6 - 64x^6 = [(2x - y)^3]^2 - (8x^3)^2
= [(2x - y)^3 + 8x^3][(2x - y)^3 - 8x^3], via difference of squares.
Using the sum and difference of cubes here gives:
(a) (2x - y)^3 + 8x^3 = (2x - y)^3 + (2x)^3
= [(2x - y) + 2x][(2x - y)^2 - (2x - y)(2x) + (2x)^2]
= (4x - y)(4x^2 - 4xy + y^2 - 4x^2 + 2xy + 4x^2)
= (4x - y)(4x^2 - 2xy + y^2).
(b) (2x - y)^3 - 8x^3 = (2x - y)^3 - (2x)^3
= [(2x - y) - 2x][(2x - y)^2 + (2x - y)(2x) + (2x)^2]
= -y(4x^2 - 4xy + y^2 + 4x^2 - 2xy + 4x^2)
= -y(12x^2 - 6xy + y^2).
Thus, the required factorization is:
-y(4x - y)(4x^2 - 2xy + y^2)(12x^2 - 6xy + y^2).
Note that the only difference between your answer and mine is the coefficient of xy in the last factor.
I hope this helps!
Source(s): Verified: http://www.wolframalpha.com/input/?i=-y%284x+-+y%2...