Factors of polynomials

Find the value of $a$.

Using your value of $a$, find $\mathrm{p}(x)$ as a product of a linear factor and a quadratic factor.

Hence show that the equation $\mathrm{p}(x) = 0$ has exactly one root.

Find the remainder when $\mathrm{p}(x)$ is divided by $x + 2$.

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

Hence solve the equation $6u^6 - u^4 - 5u^2 + 2 = 0$.

Find the values of $a$ and $b$.

Hence write $\mathrm{p}(x)$ as a product of linear factors with integer coefficients.

Using your values of $a$ and $b$, solve the equation $3\mathrm{e}^{6y} - 7\mathrm{e}^{4y} + a\mathrm{e}^{2y} + b = 0$.

Find the values of $a$ and $b$.

Show that the equation $\mathrm{p}(x) = 0$ has only one real root.

Find $\mathrm{p}(x)$ in the form $(2x-5)\mathrm{q}(x) + r$, where $\mathrm{q}(x)$ is a polynomial and $r$ is a constant.

Hence write the expression $\mathrm{p}(x) - 5$ as a product of linear factors.

Hence write down the solutions of the equation $\mathrm{p}(x) = 5$.

Show that $7a - 3b = 39$.

It is also given that when $\mathrm{p}'(x)$ is divided by $x-1$ the remainder is 1.

Find the values of $a$, $b$ and $c$.

Find the values of $a$, $b$ and $c$.

Show that the equation $\mathrm{p}(x) = 0$ has only one real root.