SOLUTION: Factor 4a^5-500a^2. I started by removing the greatest common factor. 4a^2(a^3-125) I am pretty sure that from this point it is a difference of cubes, but I am unsure as to how

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor 4a^5-500a^2. I started by removing the greatest common factor. 4a^2(a^3-125) I am pretty sure that from this point it is a difference of cubes, but I am unsure as to how       Log On


   

Question 557686: Factor 4a^5-500a^2.
I started by removing the greatest common factor.
4a^2(a^3-125)
I am pretty sure that from this point it is a difference of cubes, but I am unsure as to how I am suppose to come up with the final answer. This is due today. Thank you!
Found 2 solutions by jim_thompson5910, nerdybill:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You are correct, a%5E3-125 is a difference of cubes since a%5E3 is a perfect cube and 125=5%5E3 is also a perfect cube.


So a%5E3-125 becomes a%5E3-5%5E3 and we can use the difference of cubes formula A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29 to get


a%5E3-5%5E3=%28a-5%29%28a%5E2%2Ba%2A5%2B5%5E2%29


which simplifies to


a%5E3-5%5E3=%28a-5%29%28a%5E2%2B5a%2B25%29


So a%5E3-125=%28a-5%29%28a%5E2%2B5a%2B25%29 which means that 4a%5E5-500a%5E2 completely factors to 4a%5E2%28a-5%29%28a%5E2%2B5a%2B25%29


In other words, 4a%5E5-500a%5E2=4a%5E2%28a-5%29%28a%5E2%2B5a%2B25%29 for all values of 'a'.


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
"difference of cubes":
a^3 – b^3 = (a – b)(a^2 + ab + b^2)
.
since you have:
4a^2(a^3-125)
we can rewrite as:
4a^2(a^3-5^3)
so, now we have:
4a^2(a-5)(a^2-5a+5^2)
4a^2(a-5)(a^2-5a+25)