How do you find all the zeros of $f(x)= x^3 - 2x^2 + 5x -10$ with its multiplicities?

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Answer 1 · 2016-08-20T21:14:28

$f(x)$ has zeros $2$ and $+-sqrt(5)i$

This cubic factors by grouping then using the difference of squares identity:

$a^2-b^2=(a-b)(a+b)$

with $a=x$ and $b=sqrt(5)i$ as follows:

$x^3-2x^2+5x-10$

$=(x^3-2x^2)+(5x-10)$

$=x^2(x-2)+5(x-2)$

$=(x^2+5)(x-2)$

$=(x^2-(sqrt(5)i)^2)(x-2)$

$=(x-sqrt(5)i)(x+sqrt(5)i)(x-2)$

Hence zeros:

$+-sqrt(5)i$ and $2$

How do you find all the zeros of f(x)= x^3 - 2x^2 + 5x -10f(x)= x^3 - 2x^2 + 5x -10 with its multiplicities?