Question

Obtain all the zeroes of the polynomial $2x^4+3x^3-15x^2-24x-8$ , if two of its zeroes are $2\sqrt{2}$ and $-2\sqrt{2}$

Answer 1

If $2\sqrt{2}$ and $-2\sqrt{2}$ are zeros then, according to the factor theorem, $\left(x-2\sqrt{2}\right)\left(x+2\sqrt{2}\right)$ is a factor of the given polynomial.

$\Rightarrow \left(x^2-8\right)$ is a factor of $2x^4+3x^3-15x^2-24x-8$

Now, on dividing the given polynomial by $\left(x^2-8\right)$, we get,

$2x^2+3x+1$

On solving this quadratic polynomial, we will get the other two zeros of the given polynomial.

$\Rightarrow 2x^2+2x+x+1=0$

$\Rightarrow 2x\left(x+1\right)+\left(x+1\right)=0$

$\Rightarrow \left(x+1\right)\left(2x+1\right)=0$

Other zeros are $-1,\frac{-1}{2}$

Hence, all zeros of the given polynomial are $-1,\frac{-1}{2},2\sqrt{2},-2\sqrt{2}$.