Identical zeroes of two different nth degree polynomial equation

Asked Apr 20 '20 at 14:45

Active Apr 20 '20 at 14:45

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f(x) and g(x) are two $n^{th}$ degree polynomial equations of x. $f(x)= f_0+f_1*x^1+f_2*x^2+...+ f_n*x^n$

$g(x)= g_0+g_1*x^1+g_2*x^2+...+ g_n*x^n$

If one is to say that zeroes of both the equations are identical then can the following be said:

$\frac{f_0}{g_0}=\frac{f_1}{g_1}=\frac{f_2}{g_2}=...=\frac{f_n}{g_n}=constant$

asked Apr 20 '20 at 14:45

Sidharth

2

No, take $f=(X-1)^2(X+1)=X^3-X^2-X+1$ and $g=(X-1)(X+1)^2=X^3+X^2-X-1$. – Tuvasbien Apr 20 '20 at 14:48
Yes, if the zeros are simple. – lhf Apr 20 '20 at 14:52
Simple zero means the solution or roots don't repeat themselves as answers? For eg the equations presented by @Tuvasbien? – Sidharth Apr 20 '20 at 15:03
It means that the zeros of $f$ are not zeros of $f'$. In my example $-1$ is a simple zero of $f$ but not $1$. – Tuvasbien Apr 20 '20 at 15:09
Ohh... I think I understood the concept of simple zeroes. Is there any way to find out if an equation has all zeroes simple zeroes or not? Or do you have to factorize it ... – Sidharth Apr 20 '20 at 15:29
If you have a zero $\alpha$ of $f$, compute $f'$ and if $f'(\alpha)\neq 0$, it means that the zero $\alpha$ is simple. – Tuvasbien Apr 20 '20 at 15:35
Compute $\gcd(f,f')$. – lhf Apr 20 '20 at 18:22

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