Question

Find all the other zeroes of the polynomial $p\left(x\right)=2x^4-7x^2+19x^2-14x+30$ if two zeroes are $\sqrt{2}and-\sqrt{2}$

Answer 1

Since the two zeroes of the given polynomial $p\left(x\right)=2x^4+7x^3-19x^2-14x+30$ are $\sqrt{2}$ and $-\sqrt{2}$, therefore,

$(x-\sqrt{2})(x+\sqrt{2})=x^2-2$ is a factor of the given polynomial.

now, we divide the polynomial $p\left(x\right)=2x^4+7x^3-19x^2-14x+30$ by $x^2-2$ as shown below:

Therefore, $2x^4+7x^3-19x^2-14x+30=(x^2-2)(2x^2+7x-15)$

Now, we factorize $2x^2+7x-15$ as follows:

$2x^2+7x-15=0\Rightarrow 2x^2+10x-3x-15=0\Rightarrow 2x(x+5)-3(x+5)=0\Rightarrow (2x-3)(x+5)=0\Rightarrow 2x-3=0,x+5=0\Rightarrow 2x=3,x=-5\Rightarrow x=\frac{3}{2},x=-5$

Hence, the zeroes of the polynomial $p\left(x\right)=2x^4+7x^3-19x^2-14x+30$ are $\sqrt{2},-\sqrt{2},\frac{3}{2},-5$.