Quick questions on Factor and remainder theorems: VCE Math Methods Unit 3

For a polynomial with integer coefficients, the rational root theorem says that any rational root $p/q$ in lowest terms must have $p$ dividing the constant term and $q$ dividing the leading coefficient. So for $P(x) = 2x^3 - 3x^2 + 4x - 6$, the candidates are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 1/2, \pm 3/2$.

Each subtraction step changes signs. Write out the division carefully.

Find the remainder when $P(x) = x^3 - 4x + 7$ is divided by $(x - 2)$. [2 marks]

Fully factorise $P(x) = x^3 - 7x + 6$ over the rationals. [3 marks]

When $P(x) = x^3 + ax^2 - 3$ is divided by $(x + 1)$ the remainder is $-5$. Find $a$. [3 marks]