Question

Find all the zeros of (2x⁴ − 3x³ − 5x² + 9x − 3), it being given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$.

Answer 1

$\begin{array}{l}\text{The given polynomial is}f\left(x\right)=2x^4-3x^3-5x^2+9x-3.\\ \text{Since}\sqrt{3}\text{and}-\sqrt{3}\text{are the zeroes of}f\left(x\right),\text{it follows that each one of}(x-\sqrt{3})\text{and}(x+\sqrt{3})\text{is a factor of}f\left(x\right).\\ \\ \text{Consequently,}(x-\sqrt{3})(x+\sqrt{3})=(x^2-3)\text{is a factor of}f\left(x\right).\\ \text{On dividing}f\left(x\right)\text{by}(x^2-3)\text{, we get:}\end{array}$


$\begin{array}{l}\\ \therefore f\mathit{\left(}x\mathit{\right)}\mathit{=}\mathit{0}\\ \mathit{=}\mathit{>}2x^{\mathit{4}}\mathit{-}3x^{\mathit{3}}\mathit{-}5x^{\mathit{2}}\mathit{+}9x\mathit{-}3\mathit{=}\mathit{0}\\ \begin{array}{l}\mathit{=}\mathit{>}\mathit{(}{x}^{\mathit{2}}\mathit{-}3\mathit{)}\mathit{(}2x^{\mathit{2}}\mathit{-}3x\mathit{+}1\mathit{)}\mathit{=}\mathit{0}\\ \mathit{\text{=>}}\mathit{(}{x}^{\mathit{2}}\mathit{-}3\mathit{)}\mathit{(}2x^{\mathit{2}}\mathit{-}2x\mathit{-}x\mathit{+}1\mathit{)}\mathit{=}\mathit{0}\\ \begin{array}{l}=>(x-\sqrt{\mathrm{3}})(x+\sqrt{\mathrm{3}})(\mathrm{2}x-\mathrm{1})(x-\mathrm{1})=0\\ =>x=\sqrt{3}\mathrm{or}x=-\sqrt{3}3\mathrm{or}x=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{or}\mathit{}x=\mathrm{1}\end{array}\end{array}\\ \end{array}$