Finding constants of function $y : f(x) = x^3 + ax^2 + bx + c$

Question

The equation of a function has the form $y = f(x) = x^3 + ax^2 + bx + c$ where $a, b$ and $c$ must be determined so the function satisfies the following conditions :

• $f(2) = 0$

• $f(-1) = -12$

• $f(1) = -12$

My first thoughts were finding the properties of a derivative by using the conditions given above and possibly getting a system to solve. Nevertheless, as of now, I don't find any way how to develop this and then find the constants.

Answer 1

Hint. One may obtain $a,b,c$ by solving the linear equation system: $$\begin{cases}4a+2b+c=-8\\a-b+c=-11 \\a+b+c = -13 \end{cases}$$Can you finish it?

Answer 2

$$g(x)=f(x)-x^3$$ is a quadratic polynomial of which you know three points. You can use the Lagrangian interpolation formula

$$g(x)=(0-8)\frac{(x+1)(x-1)}{(2+1)(2-1)}+(-12+1)\frac{(x-2)(x-1)}{(-1-2)(-1-1)}+(-12-1)\frac{(x-2)(x+1)}{(1-2)(1+1)}.$$

Hence

$$f(x)=x^3+2x^2-x-14.$$