Given that ${\left(x-5\right)}^2$ is a factor of $P\left(x\right)$, write down the value of $P^{\prime }\left(5\right)$.

Write down the integer root of this equation.

Use the result from part (c)(ii) to show that $a^2-3b=0$.

Show that one of the real roots is equal to 1.

Hence or otherwise, factorize $q(x)$ as a product of linear factors.

Find $u$, $v$ and $w$ expressing your answers in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

Find the area of triangle UVW.

Show that $abc=8$.

Consider the equation $z^2+az+12=0$, where $z\in \mathrm{ℂ}$ and $a\in \mathrm{ℝ}$.

Given that $0<\alpha -\theta <\pi$, deduce that only one equilateral triangle ${\text{Z}}_1{\text{OZ}}_2$ can be formed from the point $\text{O}$ and the roots of this equation.

Write down an expression for the product of the roots, in terms of $k$.

Hence or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root.

Express $-3+\sqrt{3}\text{i}$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

Given that $q(x)$ has a factor $(x-4)$, find the value of $k$.

Given that $x^2-1$ is a factor of $f(x)$ find the value of $a$ and the value of $b$.

Factorize $f(x)$ into a product of linear factors.

Given that $q=8d^2$, find the other two real roots.

Let f(x) = x⁴ + px³ + qx + 5 where p, q are constants.

The remainder when f(x) is divided by (x + 1) is 7, and the remainder when f(x) is divided by (x − 2) is 1. Find the value of p and the value of q.

Find an expression for ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$ in terms of $p$ and $q$.

The equation $2x^2-5x+1=0$ has two real roots, $\alpha$ and $\beta$.

Consider the equation $x^2+mx+n=0$, where $m, n\in \mathrm{\mathbb{Z}}$ and which has roots $\frac{1}{{\alpha }^3}$ and $\frac{1}{{\beta }^3}$.
Without solving $2x^2-5x+1=0$, determine the values of $m$ and $n$.

Using your graph state the range of values of $c$ for which $f(x)=c$ has exactly two distinct real roots.

Prove the identity ${\left(p+q\right)}^3-3pq\left(p+q\right)\equiv p^3+q^3$.

Given that $\left(x-5\right)$ is a factor of $P\left(x\right)$, find a relationship between $a$, $b$ and $c$.

Consider the equation $z^4+a{z}^3+b{z}^2+cz+d=0$, where $a$, $b$, $c$, $d\in \mathbb{R}$ and $z\in \mathbb{C}$.

Two of the roots of the equation are log₂6 and $i\sqrt{3}$ and the sum of all the roots is 3 + log₂3.

Show that 6$a$ + $d$ + 12 = 0.

The quadratic equation $x^2-2kx+(k-1)=0$ has roots $\alpha$ and $\beta$ such that ${\alpha }^2+{\beta }^2=4$. Without solving the equation, find the possible values of the real number $k$.

Given that ${\left(x-5\right)}^2$ is a factor of $P\left(x\right)$, and that $a=2$, find the values of $b$ and $c$.

Consider the function $g\left(x\right)=x^3-3cx+d$ for $x\in \mathrm{ℝ}$ and where $c , d\in \mathrm{ℝ}$.

Find all conditions on $c$ and $d$ such that the graph of $y=g\left(x\right)$ has exactly one $x$-axis intercept, explaining your reasoning.

The cubic equation $x^3-k{x}^2+3k=0$ where $k>0$ has roots $\alpha , \beta$ and $\alpha +\beta$.

Given that $\alpha \beta =-\frac{k^2}{4}$, find the value of $k$.

The polynomial $x^4+p{x}^3+q{x}^2+rx+6$ is exactly divisible by each of $\left(x-1\right)$, $\left(x-2\right)$ and $\left(x-3\right)$.

Find the values of $p$, $q$ and $r$.