Here is a problem based on polynomials and I'm not being able to understand how to proceed. Could anyone solve it? Many thanks?

Question

How should I proceed?f(x)g(X)=0 where f(x) and g(x) are 2 degree polynomials with sum of zeroes -2 and +2 respectively. These constant term is same as well as coefficient of highest degree term. What would be the number of coincident zeroes of their product Options a.1 b.2 c.3 d.none of these

Answer 1

You are given the following

$f(x)=ax^2+bx+c$ with zeros at $p,q$. And $p+q=-2$.
$g(x)=ax^2+dx+c$ with zeros at $r,s$. And $r+s=2$.

So $f(p)=f(q)=0$
$ap^2+bp+c=0 \tag{A}$ $aq^2+bq+c=0 \tag{B}$ $(A)-(B)$
$$a(p^2-q^2)+b(p-q)=0$$ $$(p-q)[a(p+q)+b]=0$$ $$(p-q)(b-2a)=0$$ So $p=q$. And since $p+q=-2$ then $p=q=-1$. Hence $f(x)=(x+1)^2$.

The same procedure for $g(x)$ leads to $g(x)=(x-1)^2$.

Now,
$$f(x)g(x)=0$$ $$(x+1)^2(x-1)^2=0$$

So their product has four zeros, two coincident at $x=-1$ and the other two coincident at $x=1$.