SOLUTION: Can someone please help me with this question. Thank you so much Factorise each of the following expressions, that is, write them as a product. a.)8b^3a^2-4a^4b^3+16a^5b^5 b

Question 854249: Can someone please help me with this question. Thank you so much
Factorise each of the following expressions, that is, write them as a product.
a.)8b^3a^2-4a^4b^3+16a^5b^5
b.)4x^2(a+b)+4x^2(a-b)
Found 2 solutions by mananth, Theo:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!

b.)4x^2(a+b)+4x^2(a-b)

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i'll do part b first.

4x^2(a + b) + 4x^2(a - b)
simplify by performing the indicted operations to get:
4x^2*a + 4x^2*b + 4x^2*a - 4x^2*b
combine like terms to get:
8x^2*a

that should be your answer.
the 4x^2*a and the 4x^2*a add together to get 8x^2*a.
the 4x^2*b and the -4x^2*b cancel each other out.

i'll do part a next as best i can determine what they are looking for.

8*b^3*a^2-4*a^4*b^3+16*a^5*b^5
factor out a 4 to get:
4 * (2*b^3*a^2 - a^4*b^3 + 4*a^5*b^5)
factor out an a^2 to get:
4*a^2 * (2*b^3 - a^2*b^3 + 4*a^3*b^5)
factor out a b^3 to get:
4*a^2*b^3 * (2 - a^2 + 4*a^3*b^2)
you could stop there, but there is a little more factoring that can be done as follows:

reorder the terms within the parentheses to get:
4*a^2*b^3 * (4*a^3*b^2 - a^2 + 2)
factor out an a^2 from the first 2 terms in the parentheses to get:
4*a^2*b^3 * (a^2 * ((4*a*b^2 - 1) + 2)

i think that's about as far as you can go.
it helps to confirm if the final expression is equivalent to the initial expression.

a way to do that is to give random values to a and b and see if the initial expression gives you the same answer as the final expression.

i chose a = 2 and b = 3.

using those values, the initial expression of:
8*b^3*a^2-4*a^4*b^3+16*a^5*b^5 becomes:
8*3^3*2^2 - 4*2^4*3^2 + 16*2^5*3^5 which simplifies to:
864 - 1728 + 124416 which is equal to 123552.

the final expression of:
4*a^2*b^3 * ((a^2 * (4*ab^2 - 1) + 2) becomes:
(4*2^2*3^3) * ((2^2*(4*2*3^2 - 1) + 2) which becomes:
432 * (4*(71) + 2) which becomes:
432 * (284 + 2) which becomes:
432 * 286 which becomes 123552.

since both expressions give the same answer, this is a good sign that the final expression is equivalent to the original expression.

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