Question

If 2 and –2 are two zeros of the polynomial 2x⁴ – 5x³ – 11x² + 20x + 12, find all the zeros of the given polynomial.

Answer 1

Let f(x) = 2x⁴ – 5x³ – 11x² + 20x + 12

It is given that 2 and –2 are two zeroes of f(x)

Thus, f(x) is completely divisible by (x + 2) and (x – 2).

Therefore, one factor of f(x) is (x² – 4).

We get another factor of f(x) by dividing it with (x² – 4).

On division, we get the quotient 2x² – 5x – 3.

$$\begin{gathered} \Rightarrow f\left(x\right)=\left(x^2-4\right)\left(2x^2-5x-3\right) \\ =\left(x^2-4\right)\left(2x^2-6x+x-3\right) \\ =\left(x^2-4\right)\left(2x\left(x-3\right)+1\left(x-3\right)\right) \\ =\left(x^2-4\right)\left(2x+1\right)\left(x-3\right) \\ \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeroes},\mathrm{we}\mathrm{put}f\left(x\right)=0 \\ \Rightarrow \left(x^2-4\right)\left(2x+1\right)\left(x-3\right)=0 \\ \Rightarrow \left(x^2-4\right)=0\mathrm{or}\left(2x+1\right)=0\mathrm{or}\left(x-3\right)=0 \\ \Rightarrow x=\pm 2,-\frac{1}{2},3 \end{gathered}$$

Hence, all the zeroes of the polynomial f(x) are $2,-2,-\frac{1}{2}\mathrm{and}3.$