SOLUTION: factor completely 3b^2-54b+243=?

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Question 426809: factor completely
3b^2-54b+243=?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3b%5E2-54b%2B243
When factoring, always start with the Greatest Common Factor (GCF). The GCF here is 3:
3%28b%5E2-18b%2B81%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
  • b%5E2 is clearly a perfect square
  • 81 is also a perfect square, 9%5E2
  • the middle term is equal to 2*b*9
b%5E2-18b%2B81 fits the a%5E2-2ab%2Bb%5E2 pattern. Using this pattern to factor it we get:
3%28b-9%29%5E2
No further factoring can be done so we are finished.