SOLUTION: I have two problems with factorization. I tried everything, in many ways, but can't find the right answer. First: {{{(x-1)^3 + 8}}} The answer is: {{{(x+1)(x^2-4x+7)}}}

Algebra ->  Expressions-with-variables -> SOLUTION: I have two problems with factorization. I tried everything, in many ways, but can't find the right answer. First: {{{(x-1)^3 + 8}}} The answer is: {{{(x+1)(x^2-4x+7)}}}       Log On


   

Question 920810: I have two problems with factorization. I tried everything, in many ways, but can't find the right answer.
First:
%28x-1%29%5E3+%2B+8
The answer is:
%28x%2B1%29%28x%5E2-4x%2B7%29
Second:
x%5E4%2Bx%5E2y%5E2%2By%5E4
Answer:
%28x%5E2%2Bxy%2By%5E2%29%28x%5E2-xy%2By%5E2%29
Thank you!
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
1) We can factor the sum of cubes using the following formula:
x^3 + y^3 = (x+y)(x^2-xy+y^2)
we are given (x-1)^3 + 2^3
therefore we get
(x-1+2)((x-1)^2 -(x-1)2 +4))
(x+1)(x^2-2x+1 -2x +2 +4)
(x+1)(x^2 -4x +7)
2) Let a = x^2
Let b = y^2.
So lets substitute the values of a and b in for x^2 and y^2:
a^2 + ab + y^2
If you know that (a + b)^2 = a^2 + 2ab + b^2, then you can see that the two expressions are similar.
lets just add one set of "ab" to the first equation, so that we get a^2 + 2ab + b^2, which is easily simplified.
therefore, we would get:
(a + b)^2 - ab <----if we added ab, we must now subtract it.
now, just substitute the values of a and b back in:
(x^2 + y^2)^2 - x^2y^2 then apply the difference of squares here.
(x^2 + y^2) - (xy)^2
(x ^2 + y^2 + xy) (x^2 + y^2 - xy)