Question

Find the zeroes of the polynomial: $p(𝑥)={(𝑥-2)}^2-{(𝑥+2)}^2$.

Answer 1

Step 1: Form the equation

The given polynomial is $p(𝑥)={(𝑥-2)}^2-{(𝑥+2)}^2$.

We know that the zeroes of a polynomial are evaluated by equating it with zero.

$\therefore p\left(x\right)=0$

$\Rightarrow {(𝑥-2)}^2-{(𝑥+2)}^2=0$

Step 2: Solve to find the zeroes

${(𝑥-2)}^2-{(𝑥+2)}^2=0$

$\Rightarrow \left\{\left(x-2\right)+\left(x+2\right)\right\}\left\{\left(x-2\right)-\left(x+2\right)\right\}=0$ $\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]$

$\Rightarrow$ $\left(x-2+x+2\right)\left(x-2-x-2\right)=0$

$\Rightarrow$ $2x\left(-4\right)=0$

$\Rightarrow$ $-8x=0$

$\Rightarrow$ $x=0$

Hence, the zero of this polynomial is $0$.