P(x) is a cubic and when divided by a quadratic will give a result with x = degree 1.
So I set P(x) = P1(x) * (ax-b) = P2(x) * (cx-d), multiplied them out and equated coefficients of like powers.
The highest degree term and the constant term are easy. ax^3 = 2cx^3 give a = 2c and bk = -dk (and here I assumed k<>0) give b = -d.
Equating coefficients of x^2 gives d = 5c.
Plugging in these values and equating coefficients for x gives k = 30.
P1(x) = x^2 + x - 30 = (x+6)(x-5),
P2(x) = 2x^2 + 17x + 30 = (x+6)(2x+5),
(ax-b) = c(2x+5),
(cx-d) = c(x - 5),
and P(x) = c * (2x^3 + 7x^2 - 55x - 150).
Posted by xdog
on 2013-09-09 00:16:00