Use the remainder theorem to determine whether $x - 2$ is a factor of $x^3 - 7x + 6$. Provide a detailed explanation.

Given the polynomial $5x^3 - 9x^2 + 4x - 12$, demonstrate by using the remainder theorem if $x + 1$ is a factor.

How does the remainder theorem simplify the process of factoring polynomials?

If $x - 4$ is a factor of the polynomial $a(x) = x^3 - 4x^2 - 3x + c$, find the value of $c$ using the remainder theorem.

Explain how to use the remainder theorem to check if $x - 3$ is a factor of the polynomial $2x^3 - 3x^2 + 4x - 5$. Show your work step-by-step.