If 1 and $-2$ are two zeroes of the polynomial $\left(x^{3}-4 x^{2}-7 x+10\right)$, find its third zero.

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If 1 and $-2$ are two zeroes of the polynomial $\left(x^{3}-4 x^{2}-7 x+10\right)$ , find its third zero.

If 1 and $-2$ are two zeroes of the polynomial $\left(x^{3}-4 x^{2}-7 x+10\right)$ , find its third zero.

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 $\left(x^{2}+x-2\right)$ ,

f(x)=0 \Rightarrow\left(x^{2}+x-2\right)(x-5)=0 \\ \Rightarrow(x-1)(x+2)(x-5)=0 \\ \Rightarrow x=1 \text { or } x=-2 \text { or } x=5

Hence, the third zero is 5 .

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