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Sep 16, 2017

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Factor the following:
a^3 + 64 b^3

Hint: | Write 64 b^3 as a cube in order to express a^3 + 64 b^3 as a sum of cubes.
a^3 + 64 b^3 = a^3 + (4 b)^3:
a^3 + (4 b)^3

Hint: | Factor the sum of two cubes.
Factor the sum of two cubes. a^3 + (4 b)^3 = (a + 4 b) (a^2 - a×4 b + (4 b)^2):
(a + 4 b) (a^2 - 4 a b + (4 b)^2)

Hint: | Distribute exponents over products in (4 b)^2.
Multiply each exponent in 4 b by 2:
(a + 4 b) (a^2 - 4 a b + 4^2 b^2)

Hint: | Evaluate 4^2.
4^2 = 16:
(a + 4 b) (a^2 - 4 a b + 16 b^2)

Factor the following:
27 - y^12

Hint: | Factor a minus sign out of 27 - y^12.
Factor -1 out of 27 - y^12:
-(y^12 - 27)

Hint: | Express y^12 - 27 as a difference of cubes.
y^12 - 27 = (y^4)^3 - 3^3:
-(y^4)^3 - 3^3

Hint: | Factor the difference of two cubes.
Factor the difference of two cubes. (y^4)^3 - 3^3 = (y^4 - 3) ((y^4)^2 + y^4 3 + 3^2):
-(y^4 - 3) ((y^4)^2 + 3 y^4 + 3^2)

Hint: | For all positive integer exponents (a^n)^m = a^(m n). Apply this to (y^4)^2.
Multiply exponents. (y^4)^2 = y^(4×2):
-(y^4 - 3) (y^(4×2) + 3 y^4 + 3^2)

Hint: | Multiply 4 and 2 together.
4×2 = 8:
-(y^4 - 3) (y^8 + 3 y^4 + 3^2)

Hint: | Evaluate 3^2.
3^2 = 9:
-(y^4 - 3) (y^8 + 3 y^4 + 9)

Sep 16, 2017