# Tag Info

## Hot answers tagged polynomials

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### Are polynomials with the same roots identical?

No, they are not. For instance, $2x^2-2$ and $x^2-1$ have the same roots, yet they are not identical. And, depending on what you mean by "the same roots", we have that $x^2-2x+1$ and $x-1$ have the ...
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### Is the notion "If a polynomial has small coefficients (relative to the exponent), then it has small roots" true?

There exist estimates for the size of the largest root. The most general go back to the idea that $z$ is not a root of $$p(z)=a_nz^n+a_{n-1}+...+a_1z+a_0$$ if $|z|>R>0$ with an outer root ...
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### Why can you find the roots a of polynomial by factoring it?

It's not an assumption. It is a general fact that if $ab = 0$, then either $a = 0$ or $b = 0$. The roots of a polynomial are the numbers for which that polynomial evaluates to $0$, so to find the ...
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### Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$
There is no need to use Vieta's formulas. Let $$f(z)=z^4-2z^3+z^2+z-7=(z-\alpha)(z-\beta)(z-\gamma)(z-\delta).$$ Then, since $(i-a)(-i-a)=-i^2+a^2=1+a^2$, it follows that (\alpha^2+1)(\beta ^2+1)(\...