Common roots in a quadratic and a cubic polynomial

Question

Let $f(x)=x^3-3x+b$ and $g(x)=x^2+bx-3$ where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)=0$ and $g(x)=0$ have a common root? I just need a hint like from where should I start?

Answer 1

Hint: Any common root is a root of $xg(x)-f(x)$.

Answer 2

Hint: if $f(x),g(x)$ have a common root, then there exists an $x_0$ such that $f(x_0)=g(x_0)=?$

Extending this, the $?$ becomes $0$. As demonstrated in other answers, a root $x=0$ can be added to make the two functions have the same degree, with $xg(x)-f(x)=0$, and then the problem becomes quadratic at worst.

Answer 3

Let $y$ be the common root

$$0=y^3-3y+b=y(y^2+by-3)-b(y^2+by-3)=3b-b^2y$$

Either $b=0,$ or $y=\dfrac3b$

Put this value of $y$ in $y^2+by-3=0$