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In the marking scheme they somehow manipulated a cubic to retrieve one of the factors needed to answer the question: enter image description here

My question is: How can it be known to do this baring in mind there is three roots and the others had many decimals? Its like they pulled it from nowhere.

ervx
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DevinJC
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1 Answers1

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Let $p(k)=k^{3}+3k^{2}-36k+52$. Since $p(2)=0$, the Factor Theorem

gives that $k-2$ is a factor of $p(k)$.

You then divide the polynomial $p(k)$ by $k-2$ to obtain the quotient $k^{2}+5k-26$. Thus, $p(k)=(k-2)(k^{2}+5k-26)$. This is the correct factorization.

ervx
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  • Thanks. I do know about the factor theorem but it just seems strange that they would've of wanted the reader to do all that when the question was about matrices originally lol. – DevinJC Sep 16 '18 at 16:43
  • Well, if you're finding eigenvalues, you need to know how to do this. – Sean Roberson Sep 16 '18 at 16:54
  • @DevinJC You haven‘t shown us the matrix for which this is the characteristic polynomial. It might be in some way obvious that one of its eigenvalues is $2$. – amd Sep 17 '18 at 07:16
  • @amd Haven't learnt about them at all so I didn't think of posting the original matrix sorry. – DevinJC Sep 17 '18 at 10:14