When looking for zeros of polynomials there are only few fast methods to find them quickly. e.g you have the function: $x^3-3x^2-4x+12=0$ What you can do is you find all the integers that are divisors of the absolute term and than put them in and find out if you got zeros. My question is now, if I use this method how can I determine if I got a double zero at any of those points? And another question would be if there are any other fast methods to find zeros.
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2A double zero is also a zero of the derivative and you know where the zeroes of $3x^2-6x-4$ lie. – Jack D'Aurizio Jul 08 '17 at 16:59
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This could help or apply rational root theorem as mentioned in link – Atul Mishra Jul 08 '17 at 16:59
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There are two ways. Say you know that $1$ is a root of $x^3 - 3x + 2$. Then you can calculate the remainder:
$$ \frac{x^3 - 3x + 2}{x - 1} = x^2 + x - 2$$
and check whether or not $1$ is a root of that remainder.
Alternatively, you can calculate the derivative
$$ \frac{d}{dx} ( x^3 - 3x + 2 ) = 3x^2 - 3 $$
and check whether or not $1$ is a root of the derivative.
Note that a root $r$ is a double root of $f(x)$ if and only if $r$ is a root of $\frac{f(x)}{x - r}$, and also $r$ is a double root of $f(x)$ if and only if $r$ is a root of $f'(x)$.
Sera Gunn
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Hint:
In this case we have:
$$ x^3-3x^2-4x+12=x^2(x-3)-4(x-3)=(x-3)(x^2-4)=(x-3)(x-2)(x+2)=0 $$
Emilio Novati
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