SOLUTION: I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!!! I would deeply appreciate it!! Solving Polynomial Eq

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Question 166923This question is from textbook Algebra and Trigonometry Structure and Method book 2
: I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!!! I would deeply appreciate it!!
Solving Polynomial Equations
Solve. Identify all double roots.
(s-6)^3=27(s-8) This question is from textbook Algebra and Trigonometry Structure and Method book 2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28s-6%29%5E3=27%28s-8%29 Start with the given equation


s%5E3-18s%5E2%2B108s-216+=27%28s-8%29 Expand %28s-6%29%5E3 to get s%5E3-18s%5E2%2B108s-216. Hint: Use Pascal's Triangle to expand


s%5E3-18s%5E2%2B108s-216+=+27s+-+216 Distribute


s%5E3-18s%5E2%2B108s+=+27s Add 216 to both sides. Notice that they cancel out and go away.


s%5E3-18s%5E2%2B108s+-+27s+=+0 Subtract 27s from both sides.


s%5E3-18s%5E2%2B+81s+=+0 Combine like terms.


s%28s%5E2-18s%2B81%29=0 Factor out the GCF "s" from the left side


s%28s-9%29%28s-9%29=0 Factor s%5E2-18s%2B81 to get %28s-9%29%28s-9%29


Now set each factor equal to zero:


s=0, s-9=0 or s-9=0


s=0, s=9 or s=9 Now solve for s in each case


Since we have a repeating answer, our answers are s=0 or s=9


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Answer:

So the solutions are s=0 or s=9 where s=9 is a double root (ie has a multiplicity of 2)