If a polynomial $x^4+5x^3+4x^2-10x-12$ has two zeroes as $-2$ and $-3$, then find the other zeroes.

Given: A polynomial $x^4+5x^3+4x^2-10x-12$ has two zeroes as $-2$ and $-3$.

To do: To find the other zeroes.

Solution: 

As given polynomial $x^4+5x^3+4x^2-10x-12$ has two zeroes as $-2$ and $-3$.

Then, $( x+2)$ and $( x+3)$ are the factors of the polynomial $x^4+5x^3+4x^2-10x-12$.

Let, $g( x)=( x+2)( x+3)=x^2+3x+2x+6$

$\Rightarrow g( x)=x^2+5x+6$

$\because ( x+2)$ and $( x+3)$ are the factors of the polynomial $x^4+5x^3+4x^2-10x-12$. then, On dividing the polynomial $x^4+5x^3+4x^2-10x-12$ by $g( x)=x^2+5x+6$.

We get the quotient $x^2-2$. Thus $x^2-2$ is also a factor of the polynomial.

$\Rightarrow x^2-2=0$

$\Rightarrow x^2=2$

$\Rightarrow x=\pm\sqrt{2}$

Thus, $\sqrt{2}$ and $-\sqrt{2}$ are the other zeroes of the polynomial $x^4+5x^3+4x^2-10x-12$.

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Updated on: 10-Oct-2022

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