Question

If 2 and $\frac{-1}{3}$ are the zeros of the polynomial 3x³ − 2x² − 7x − 2, find the third zero of the polynomial.

Answer 1

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeros}\mathrm{of}\text{a}\mathrm{polynomial},\mathrm{we}\mathrm{equate}\mathrm{th}\text{e}\mathrm{polynomial}\mathrm{to}0. \\ 3x^3-2x^2-7x-2=0 \\ \Rightarrow 3x^3-6x^2+4x^2-8x+x-2=0 \\ \Rightarrow 3x^2\left(x-2\right)+4x\left(x-2\right)+1\left(x-2\right)=0 \\ \Rightarrow \left(x-2\right)\left(3x^2+4x+1\right)=0 \\ \Rightarrow \left(x-2\right)\left(3x^2+3x+x+1\right)=0 \\ \Rightarrow \left(x-2\right)\left[3x\left(x+1\right)+1\left(x+1\right)\right]=0 \\ \Rightarrow \left(x-2\right)\left(x+1\right)\left(3x+1\right)=0 \\ \Rightarrow x-2=0\mathrm{or}x+1=0\mathrm{or}3x+1=0 \\ \Rightarrow x=2\mathrm{or}-1\mathrm{or}\frac{-1}{3} \\ \mathrm{As}2\mathrm{and}\frac{-1}{3}\mathrm{are}\text{the}\mathrm{given}\mathrm{zeros},\mathrm{the}\mathrm{third}\mathrm{zero}\text{is}-1. \end{gathered}$$