Question:
If 1 and $-2$ are two zeros of the polynomial $\left(x^{3}-4 x^{2}-7 x+10\right)$, find its third zero.
Solution:
Let $f(x)=x^{3}-4 x^{2}-7 x+10$
Since 1 and $-2$ are the zeroes of $f(x)$, it follows that each one of $(x-1)$ and
$(x+2)$ is a factor of $f(x)$
Consequently, $(x-1)(x+2)=\left(x^{2}+x-2\right)$ is a factor of $f(x)$.
On dividing $f(x)$ by $\left(x^{2}+x-2\right)$, we get:
$f(x)=0=>\left(x^{2}+x-2\right)(x-5)=0$
$=>(x-1)(x+2)(x-5)=0$
$\Rightarrow x=1$ or $x=-2$ or $x=5$
Hence, the third zero is 5 .
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