Roots of Polynomials

Let $p(x) = ax^{3} + bx - c$ Let $g(x) = x^{2} + bx + c$ Now, when forming the polynomial g(x) , the constant was k = 1 But putting k = a will not change g(x)'s roots => $g(x) = a(x^{2} + bx + c )$ => $g(x) = ax^{2} + abx + ac$

Now dividing p(x) by g(x) will give us $q(x) = x-b$ And p(x) divided by q(x) will give us g(x). But while taking the quotient we find that we get $g(x) = ax^{2} + abx + b(ab + 1)$ and remainder = $b^{2}(ab +1) - c = 0$ Equating the two expressions of g(x) we get $b(ab+1)=ac -----(1)$ From remainder; $b^{2}(ab+1) - c = 0$ => $b×b(ab+1) = c$ => $b×ac = c from (1)$ => $ab =1$ Q.E.D