SOLUTION: factor 625s^6-5f^6 the correct factorization is?

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Question 426811: factor
625s^6-5f^6
the correct factorization is?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
625s%5E6-5f%5E6
When factoring, always start with the Greatest Common Factor (GCF). The GCF here is 5:
5%28125s%5E6-f%5E6%29
Next look to the other methods of factoring:
  • Patterns
    • a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29
    • a%5E2%2B2ab%2Bb%5E2+=+%28a%2Bb%29%28a%2Bb%29+=+%28a%2Bb%29%5E2
    • a%5E2-2ab%2Bb%5E2+=+%28a-b%29%28a-b%29+=+%28a-b%29%5E2
    • a%5E3%2Bb%5E3+=+%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29
    • a%5E3-b%5E3+=+%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29
  • Trinomials
  • Factoring by grouping
  • Factoring by trial and error of the rational roots

Since 125+=+5%5E3 and since exponents that are multiples of 3 (like 6's) are perfect cubes, the second factor fits the pattern for difference of cubes (the last one above) with the "a" being 5s%5E2 and the "b" being r%5E2, Using this pattern to factor it we get:
5%285s%5E2-s%5E2%29%28%285s%5E2%29%5E2%2B%285s%5E2%29%28r%5E2%29%2B%28r%5E2%29%5E2%29
which simplifies to:
5%285s%5E2-s%5E2%29%2825s%5E4%2B5s%5E2r%5E2%2Br%5E4%29
And try as we might (and we should) none of these factors will factor further so we are finished.