Answer to Question #236234 in Math for luka

Question

If X^2+ bX +c and X^2+ dX + e have a common factor ( X – p ) show that p =e-c/b-d

Answer 1

If $x^2+ bx +c$ has a factor $(x-p),$ then

$$p^2+bp+c=0$$

If $x^2+ dx +e$ has a factor $(x-p),$ then

$$p^2+dp+e=0$$

Hence

$$p^2+bp+c=p^2+dp+e$$

$$bp+c=dp+e$$

$$(b-d)p=e-c$$

$$p=\dfrac{e-c}{b-d}, b\not=d$$

If $b=d,$ then $c=e$ and the quadratic polynomials are the same for $p\in\R.$