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Let the roots of the polynomial x^2 + 7x - 2 be alpha and beta.  Evaluate alpha^2*beta + alpha*beta^2.

 Jul 23, 2022
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P(x) = x^2 + 7x - 2 = (x - a)(x - b)

 

Q(x) = (x - a^2)(x - b^2) 

Q(x^2) = (x - a)(x - b)(x + a)(x + b) using difference of squares.

= P(x) * P(-x) 

 

(x^2 -2 + 7x)(x^2 - 2 - 7x) = Q(x^2)

= (x^2 - 2)^2 - 49x^2 

 

From here, simplify, and for the x^2 terms, make them x which will give you Q(x). Use vieta's to find the sum of the roots and you will be done. :)

 Jul 23, 2022

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