Question

If one zero of the polynomial p(x) = x3 − 6x² + 11x − 6 is 3, find the other two zeroes.

Answer 1

$$\begin{gathered} \mathrm{Given}:p\left(x\right)=x^3-6x^2+11x-6\mathrm{}\mathrm{and}\mathrm{its}\mathrm{factor},x+3 \\ \mathrm{Let}\mathrm{us}\mathrm{divide}p\left(x\right)\mathrm{by}(x-3). \end{gathered}$$

$$\begin{gathered} \mathrm{Here},x^3-6x^2+11x-6=\left(x-3\right)\left(x^2-3x+2\right) \\ =\left(x-3\right)\left[x^2-\left(2+1\right)x+2\right] \\ =\left(x-3\right)(x^2-2x-x+2) \\ =\left(x-3\right)[x\left(x-2\right)-1(x-2\left)\right] \\ =\left(x-3\right)\left(x-1\right)\left(x-2\right) \\ \therefore \mathrm{The}\mathrm{}\mathrm{other}\mathrm{}\mathrm{two}\mathrm{}\mathrm{zeroes}\mathrm{}\mathrm{are}1\mathrm{and}2. \end{gathered}$$