Author Topic: Polynomial with roots which are the squares of another polynomials roots (Read 1030 times) Tweet Share

frog0101 · « **on:** October 21, 2018, 08:33:23 pm »

Hi,
For a question such as 2016 HSC Q15 (a), given the polynomial,
$x^3-3x+1=0 \textit{which has roots }\alpha ,\beta ,\gamma$ , find a cubic equation with integer coefficients that has roots $\alpha ^2, \beta ^2, \gamma ^2$ .

If I approach the question:
$\alpha _i^{2}=t, \alpha _i \textit{ represents any root of } <br />\displaystyle P(x)=x^3-3x+1$

$\textit{Why does }\alpha _i=\sqrt{t}, \textit{instead of }\alpha _i=\pm \sqrt{t} \textit{ ?}$

RuiAce · « **Reply #1 on:** October 21, 2018, 09:41:55 pm »

Note how the question says "find a cubic equation" and so on. Because we only want one cubic equation, which is what they'll typically always do, we can just take the positive root for convenience.

It turns out that there's gonna be more than one cubic equation that works in general (in fact, up to 8 of them). If we wanted to "find all cubic equations", then the computations would be more demanding and require the cases that you've considered.

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Author Topic: Polynomial with roots which are the squares of another polynomials roots (Read 1030 times) Tweet Share

frog0101

Polynomial with roots which are the squares of another polynomials roots

RuiAce

Re: Polynomial with roots which are the squares of another polynomials roots

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