Question

Verify that $-5$ and $7$ are the zeroes of the polynomial $p\left(x\right)=\left(x+5\right)\left(x-7\right)$.

Answer 1

Find the value of polynomial $p\left(x\right)=\left(x+5\right)\left(x-7\right)$ at $x=-5\text{and}7$:

Defintion:

Any real number $\text{'}a\text{'}$ is root or zero of a polynomial $p\left(x\right)$ if and only if $p\left(a\right)=0$.

$\begin{array}{rcl}& \Rightarrow & p\left(-5\right)=\left(-5+5\right)\left(-5-7\right)\\ & =& \left(0\right)\left(-12\right)\\ & =& 0\end{array}$
$\begin{array}{rcl}& \Rightarrow & p\left(7\right)=\left(7+5\right)\left(7-7\right)\\ & =& \left(12\right)\left(0\right)\\ & =& 0\end{array}$

By using the above-mentioned definition, $x=-5\text{and}7$ is a zero of polynomial $p\left(x\right)=\left(x+5\right)\left(x-7\right)$.

Hence, it is verified.