Question

The most general cubic (third-degree) equation with rational coefficients can be written as x³ + ax² + bx + C -0. Prove that if we replace x by X-a/3 and simplify, we end up with an equation that doesn't have an X2 term, that is, an equation of the form X³ + PX + 9 = 0.

Answer 1

By substituting X = x - a/3 into the given cubic equation and simplifying, we can obtain an equation without an X^2 term.

Step-by-step explanation:

To prove that if we replace x by X-a/3 and simplify, we end up with an equation that doesn't have an X^2 term, we can start by substituting X = x - a/3 into the given equation x^3 + ax^2 + bx + c = 0.

After simplifying, we obtain (x - a/3)^3 + a(x - a/3)^2 + b(x - a/3) + c = 0.

Finally, expanding and simplifying this equation will lead to an equation of the form X^3 + PX + Q = 0, where P and Q are constants.