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Given the quadratic polynomial $ax^2 + bx + c$, find a new polynomial with coefficients expressed in terms $a$, $b$ and $c$ such that the product and the sum of its zeros will be the sum and the product, respectively, of the zeros of the original polynomial.

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Recall that for the quadratic polynomial $ax^2 + bx + c$, the sum of the roots is $-\dfrac{b}a$ while the product of the roots is $\dfrac{c}a$. Now you want a quadratic whose sum is $\dfrac{c}a$ and the product is $-\dfrac{b}a$. Hence the quadratic you are after is $$\alpha \left(y^2 - \dfrac{c}ay - \dfrac{b}a\right)$$ where $\alpha$ is some constant. Since you want one such polynomial, we can choose $\alpha = a$, and this gives us $$ay^2 - cy - b$$

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