Can any cubic polynomial be transformed into canonical form?

Question

Can any cubic polynomial be transformed from $Ax^3+Bx^2+Cx+D$ to $a(b(x-h))^3 + k$?

For example, how could $x^3+\frac{3x^2}{2}+\frac{x}{2}$ be transformed?

Answer 1

If you're asking whether a cubic polynomial $ A x^3 + B x^2 + C x + D $ can be put in the form $ A (x - k)^3 + h $, then the answer is no. While in the quadratic case it is always possible to convert $A x^2 + B x + C $ into the form $A (x - k)^2 + h $

However, it is possible to eliminate the $x^2$ term, by writing

$ A x^3 + B x^2 + C x + D = A (x - k)^3 + C_1 (x - k) + h$

Comparing the forms on the left and the right, we get

$ B = A (-3 k) $

$ C = A (9 k^2) + C_1 $

$ D = h - A k^3 - C_1 k $

From the first, one can find $k$, then from the second equation, one finds $C_1$, and finally from the third equation, one finds $h$.