SOLUTION: one of the factors of the expression (2a+5b)^3+(2a-5b)^3 would be? a)4a b)10b c)2a+5b d)2a-5b also give me the solution!thankyou!!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: one of the factors of the expression (2a+5b)^3+(2a-5b)^3 would be? a)4a b)10b c)2a+5b d)2a-5b also give me the solution!thankyou!!      Log On


   

Question 714736: one of the factors of the expression (2a+5b)^3+(2a-5b)^3 would be?
a)4a b)10b
c)2a+5b d)2a-5b
also give me the solution!thankyou!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%282a%2B5b%29%5E3%2B%282a-5b%29%5E3
As weird as this may look this expression, in essence, is a sum of cubes. And we have a factoring pattern for a sum of cubes:
p%5E3%2Bq%5E3=%28p%2Bq%29%28p%5E2-pq%2Bq%5E2%29
So we can use this pattern (with the "p" being "2a+5b" and the "q" being "2a=5b") to factor %282a%2B5b%29%5E3%2B%282a-5b%29%5E3:

Now we simplify the factors. In the first factor the 5b's cancel out leaving 2a+2a or 4a. So the multiple-choice answer is A.

To simplify %28%282a%2B5b%29%5E2-%282a%2B5b%29%282a-5b%29%2B%282a-5b%29%5E2%29 more easily we can use the following patterns:
  • %28p%2Bq%29%5E2+=+p%5E2%2B2pq%2Bq%5E2 on %282a%2B5b%29%5E2
  • %28p%2Bq%29%28p-q%29+=+p%5E2-q%5E2 on %282a%2B5b%29%282a-5b%29
  • %28p-q%29%5E2+=+p%5E2-2pq%2Bq%5E2 on %282a-5b%29%5E2%29
Using these patterns we get:

Simplifying...
%28%284a%5E2%2B20ab%2B25b%5E2%29-%284a%5E2-25b%5E2%29%2B%284a%5E2-20ab%2B25b%5E2%29%29
The 20ab's cancel out. And two of the 4a%5E2's cancel out. Adding the other like terms we get:
%284a%5E2%2B75b%5E2%29

Altogether
%282a%2B5b%29%5E3%2B%282a-5b%29%5E3
factors into:
4a%284a%5E2%2B75b%5E2%29