Question

Find all zeroes of the polynomial 3x³ + 10x² – 9x – 4, if one of its zeroes is 1.

Answer 1

Let f(x) = 3x³ + 10x² – 9x – 4

It is given that one of its zeroes is 1

Therefore, one factor of f(x) is (x – 1).

We get another factor of f(x) by dividing it with (x – 1).

On division, we get the quotient 3x² + 13x + 4.

$$\begin{gathered} \Rightarrow f\left(x\right)=\left(x-1\right)\left(3x^2+13x+4\right) \\ =\left(x-1\right)\left(3x^2+12x+x+4\right) \\ =\left(x-1\right)\left(3x\left(x+4\right)+1\left(x+4\right)\right) \\ =\left(x-1\right)\left(3x+1\right)\left(x+4\right) \\ \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeroes},\mathrm{we}\mathrm{put}f\left(x\right)=0 \\ \Rightarrow \left(x-1\right)\left(3x+1\right)\left(x+4\right)=0 \\ \Rightarrow \left(x-1\right)=0\mathrm{or}\left(3x+1\right)=0\mathrm{or}\left(x+4\right)=0 \\ \Rightarrow x=1,-\frac{1}{3},-4 \end{gathered}$$

Hence, all the zeroes of the polynomial f(x) are $1,-\frac{1}{3}\mathrm{and}-4.$