Find the zeros of the polynomial

Question:

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

 

Solution:

$f(x)=x^{2}+x-p(p+1)$

By adding and subtracting px, we get

$f(x)=x^{2}+p x+x-p x-p(p+1)$

$=x^{2}+(p+1) x-p x-p(p+1)$

$=x[x+(p+1)]-p[x+(p+1)]$

$=[x+(p+1)](x-p)$

$f(x)=0$

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

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

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

So, the zeros of f(x) are −(p + 1) and p.

 

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