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 $p\in\R.$