Question

Obtain all other zeros of (x⁴ + 4x³ − 2x² − 20x − 15) if two of its zeros are $\sqrt{5}$ and $-\sqrt{5}$.

Answer 1

$\begin{array}{l}\text{The given polynomial is}f\left(x\right)=x^4+4x^3-2x^2-20x-15.\\ \text{Since}(x-\sqrt{5})\text{and}(x+\sqrt{5})\text{are the zeroes of}f\left(x\right),\text{it follows that each one of}(x-\sqrt{5})\text{and}(x+\sqrt{5})\text{is a factor of}f\left(x\right).\\ \\ \text{Consequently,}(x-\sqrt{5})(x+\sqrt{5})=(x^2-5)\text{is a factor of}f\left(x\right).\\ \\ \text{On dividing}f\left(x\right)\text{by}(x^2-5)\text{, we get:}\\ \end{array}$


$\begin{array}{l}\therefore f\left(x\right)=0\\ =>x^4+4x^3-7x^2-20x-15=0\\ \begin{array}{l}=>(x^2-5)(x^2+4x+3)=0\\ =>(x-\sqrt{5})(x+\sqrt{5})(x+1)(x+3)=0\\ =>x=\sqrt{5}\text{or}x=-\sqrt{5}\mathrm{or}x=-1\text{or}x=-3\\ \text{Hence, all the zeros are}\sqrt{5},-\sqrt{5},-1\mathrm{and}-3.\end{array}\\ \\ \end{array}$