Given polynomials $p(x)$ and $q(x)$. If $z_1, z_2, z_3, z_4$ are roots of $q(x)$, then find the value of $p(z_1)+p(z_2) +p(z_3) +p(z_4)$.

Asked Mar 21 '22 at 09:19

Active Mar 21 '22 at 09:21

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Let $p(x)=x^6-x^5-x^3-x^2-x$ and $q(x)=x^4-x^3-x^2-1$. If $z_1, z_2, z_3, z_4$ are roots of $q(x)$, then find the value of $p(z_1)+p(z_2) +p(z_3) +p(z_4)$.

My attempt:
Dividing $p(x)$ by $q(x)$ gives $$p(x)=(x^2+1) \cdot q(x) +(x^2-x+1)$$ So, if $z$ is a root of $q(x)$ then $$p(z) = z^2-z+1$$

I also know that $-1$ is a root of $q(x)$. But idk how to get the value of $p(x)$ for the other roots. Any help?

edited Mar 21 '22 at 09:21

asked Mar 21 '22 at 09:19

Agung Izzul Haq

3

So you need $\sum_{i = 1}^4 {z_i^2 } - \sum_{i = 1}^4 {z_i } + 4$. Do you know Newton's identities? – Gary Mar 21 '22 at 09:23
1

Ah, I see. I know about it. Thanks. – Agung Izzul Haq Mar 21 '22 at 09:34

Given polynomials $p(x)$ and $q(x)$. If $z_1, z_2, z_3, z_4$ are roots of $q(x)$, then find the value of $p(z_1)+p(z_2) +p(z_3) +p(z_4)$.

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