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Ok so I was going through a limit question, and I am only interested in one aspect of the question it is mention that

Let f(x) be a polynomial which is satisfying

${f(a)}^{2} + {f'(a)}^{2} = 0$, then that means $f(a) = 0 = f'(a)$. This means that x = a is a root of $f'(x)$, now this is the part i don't understand whatsoever, because in my book its written that $(x-a)^2$ is a factor of f(x). My question is why is it required to take a square of the root $(x-a)$? I'd like to see an example to help me understand this please

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Since $a$ is a root of $f(x)$, you can write $f(x)$ as $(x-a)p(x)$ for some polynomial $p(x)$. Suppose that $a$ is not a root of $p(x)$. Then $f'(x)=p(x)+(x-a)p'(x)$ and therefore $f'(a)=p(a)\ne0$. But we are assuming that $f'(a)=0$. So, $a$ has to be a root of $p(x)$. So, $x-a\mid p(x)$, and therefore $(x-a)^2\mid f(x)$.

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