Question

Find the zeros of the polynomial $x^2+x-p(p+1)$.

Answer 1

Let, $f\left(x\right)=x^2+x-p(p+1)$

Using Middle Term Splitting, we have to find two numbers whose sum is $1$ and product is $-p(p+1)$.

Thus, the two numbers are $-p$ and $p+1$

$\Rightarrow x^2-px+(p+1)x-p(p+1)=0$

$\Rightarrow x(x-p)+(p+1)(x-p)=0$

$\Rightarrow (x+p+1)(x-p)=0$

$\Rightarrow x=-(p+1),p$

Therefore, zeroes of the given polynomial are $-(p+1)$ and $p$.