Factors of polynomials

Given that $x = 2$ and $x = -1$ are roots of the equation $\mathrm{p}(x) = 0$, find $a$ and $b$.

Solve the equation $\mathrm{p}(x) = 0$.

Show that $x + 3$ is a factor of $-12 + 23x + 3x^2 - 2x^3$.

The curve $y = -5 + 33x + 3x^2 - 2x^3$ and the line $y = 10x + 7$ intersect at three points, $A$, $B$ and $C$. These points are such that the $x$-coordinate of $A$ has the least value and the $x$-coordinate of $C$ has the greatest value. Show that $B$ is the mid-point of $AC$.

Find the remainder when $\mathrm{p}(x)$ is divided by $x + 1$.

Show that $x + 2$ is a factor of $\mathrm{p}(x)$.

Write $\mathrm{p}(x)$ as a product of linear factors.

Given that $x - 2$ is a factor of $ax^3 - 12x^2 + 5x + 6$, use the factor theorem to show that $a = 4$.

Showing all your working, factorise $4x^3 - 12x^2 + 5x + 6$ and hence solve $4x^3 - 12x^2 + 5x + 6 = 0$.

The cubic equation $x^3 + ax^2 + bx - 40 = 0$ has three positive integer roots. Two of the roots are 2 and 4. Find the other root and the value of each of the integers $a$ and $b$.

Do not use a calculator in this question.

Solve the equation $x^3 - 5x^2 - 46x - 40 = 0$ given that it has three integer roots, only one of which is positive.