Question

If two zeroes of the polynomial p(x) = 2x⁴ − 3x³ − 3x² + 6x − 2 are $\sqrt{2}$ and $-\sqrt{2}$, find its other two zeroes.

Answer 1

$$\begin{gathered} \mathrm{Given}:\mathrm{p}\left(\mathrm{x}\right)=2x^{\mathit{4}}-3x^3-3x^2+6x-2\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{}\mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{e}\mathrm{s},\sqrt{2}\mathrm{and}\mathrm{}-\sqrt{2} \\ \mathrm{So},\mathrm{}\mathrm{the}\mathrm{polynomial}\mathrm{is}\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)=x^2-2. \\ \mathrm{Let}\mathrm{us}\mathrm{divide}\mathrm{}p\left(x\right)\mathit{}\mathrm{b}\mathrm{y}\mathrm{}\left(x^2-2\right).\mathrm{}\mathrm{}\mathrm{} \end{gathered}$$

$$\begin{gathered} \mathrm{Here},2{\mathrm{x}}^4-3{\mathrm{x}}^3-3{\mathrm{x}}^2+6\mathrm{x}-2=\left(x^{\mathit{2}}\mathit{-}\mathit{2}\right)\left(\mathit{2}{x}^{\mathit{2}}\mathit{-}\mathit{3}x\mathit{+}\mathit{1}\right) \\ \mathit{=}\left(x^{\mathit{2}}\mathit{-}\mathit{2}\right)\left[\mathit{2}{x}^{\mathit{2}}\mathit{-}\left(\mathit{2}\mathit{+}\mathit{1}\right)x\mathit{+}\mathit{1}\right] \\ \mathit{=}\left(x^{\mathit{2}}\mathit{-}\mathit{2}\right)\mathit{(}\mathit{2}{x}^{\mathit{2}}\mathit{-}\mathit{2}x\mathit{-}x\mathit{+}\mathit{1}\mathit{)} \\ =(x^2-2)\left[\right(2x(x-1)-1(x-1)] \\ \mathit{=}\left(x^{\mathit{2}}\mathit{-}\mathit{2}\right)\left(\mathit{2}x\mathit{-}\mathit{1}\right)\left(x\mathit{-}\mathit{1}\right) \\ \therefore \mathrm{Th}\mathrm{e}\mathrm{}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{}\mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{e}\mathrm{s}\mathrm{}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{}\frac{1}{2}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}1. \end{gathered}$$