Polynomial functions + Sum & Products of roots: notes and practice questions

(a) Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

(b) By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8 \cos^4\theta - 8 \cos^2\theta + 1$.

(c) Hence, find the four distinct roots of the equation $8x^4 - 8x^2 + 1 = 0$, expressing them in the form $\cos(\alpha)$ where $0 < \alpha < \pi$.

(d) By considering the roots of the equation in part (c), or otherwise, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

Consider the function $f$ defined by $f(x) = x^3 - 6x^2 + 8x$.

Find the $x$-intercepts of the graph of $y=f(x)$.

The graph of $y=f(x)$ for $0 \le x \le 4$ is shown below. The graph encloses two regions with the $x$-axis, shaded in the diagram.

Find the total area of the shaded regions.

The total surface area of a closed right cylinder is 8, equal to the total shaded area found in part (b). The cylinder has a height of $\frac{4-\pi}{\pi}$.

Find the radius, $r$, of the cylinder.

Hence, find the volume of the cylinder.

Consider the complex number $w = 1 - i$.

By finding the modulus and argument of $w$, show that $w = \sqrt{2}e^{-i\frac{\pi}{4}}$.

Find the smallest positive integer $n$ such that $w^n$ is a real number.

Find the value of $w^n$ when $n$ takes the value found in part (b)(i).

Consider the equation $z^3 - 2z + 4 = 0$, where $z \in \mathbb{C}$.

Given that $w=1-i$ is a root of this equation, find the other roots.

By using a suitable transformation from $z$ to $v$, or otherwise, find the roots of the equation $4v^3 - 2v^2 + 1 = 0$, where $v \in \mathbb{C}$.

Consider the equation $z^2 = -4z^*$, where $z \in \mathbb{C}, z \neq 0$.

By expressing $z$ in the form $a + bi$, find the roots of the equation.

Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$.

Hence, show that $\theta = \frac{\pi}{8}$ and $\theta = \frac{3\pi}{8}$ are solutions of the equation $8\cos^4\theta - 8\cos^2\theta + 1 = 0$.

Hence, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

A function $f$ is defined by $f(x) = x^3 - 6x^2 + cx + d$, where $c, d \in \mathbb{R}$. The equation $f(x)=0$ has three real roots, $\alpha, \beta$ and $\gamma$.

(a) Write down the value of $\alpha + \beta + \gamma$.

A polynomial $P$ is defined by $P(z) = z^5 - 8z^4 + pz^3 + qz^2 + rz - 12$, where $p, q, r \in \mathbb{R}$. The roots of $f(x)=0$ are also roots of the equation $P(z)=0$. It is given that $z=1-i$ is a root of $P(z)=0$.

(b) Find the other non-real root of $P(z)=0$, giving a reason for your answer.

(c) Find the value of $\alpha\beta\gamma$.

(d) It is given that the roots $\alpha, \beta, \gamma$ form an arithmetic progression. Find the values of $\alpha, \beta$ and $\gamma$.

Prove the identity $(x + y)^3 - 3xy(x + y) = x^3 + y^3$.

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

Consider the equation $x^2 + mx + n = 0$, where $m, n \in \mathbb{Z}$ and which has roots $\alpha^3$ and $\beta^3$.

Without solving $x^2 - 5x + 2 = 0$, determine the values of $m$ and $n$.

The graph of the function $f(x) = x^3 - 5x^2 + 8x$ and the line $L$ with equation $y=4x$ are shown in the diagram below. The graphs intersect at the origin O, and at points A and B.

(a) Find the coordinates of A and B.

(b) The region enclosed by the graph of $f(x)$ and the line $L$ is composed of two smaller regions. Find the total area of these two enclosed regions.

(a) Find the binomial expansion of $(2x - \frac{3}{x})^3$ in ascending powers of $x$.

(b) Hence, or otherwise, find the term independent of $x$ in the binomial expansion of $(5+x)(2x - \frac{3}{x})^3$.

Consider the complex number $z_1 = 1 - i\sqrt{3}$.

(a) (i) Express $z_1$ in modulus-argument form.

(a) (ii) Find the smallest positive integer $n$ for which $z_1^n$ is a real number.

Consider the equation $(z+i)^5 - (z-i)^5 = 0$, where $z \in \mathbb{C}$.

(b) Show that the roots of the equation are given by $z = \cot\left(\frac{k\pi}{5}\right)$ for $k=1, 2, 3, 4$.

(c) (i) By using the binomial expansion, show that the equation in part (b) can be written as $5z^4 - 10z^2 + 1 = 0$.

(c) (ii) Let the roots of the equation in (c)(i) be $z_1, z_2, z_3, z_4$. Without finding the roots, show that $\sum_{j=1}^4 z_j^2 = 4$.

(d) Hence, find the exact value of $\cot^2\left(\frac{\pi}{5}\right)$.

A polynomial is given by $P(x) = 2x^4 + (k-1)x^3 - kx^2 + 5x - (k+3)$, where $k \in \mathbb{R}$.

(a) The sum of the roots of the equation $P(x)=0$ is 3. Find the value of $k$.

(b) Hence, find the product of the roots of the equation $P(x)=0$.

Find all the solutions to the equation $\cos 5\theta = -\cos 4\theta$ in the interval $0 \le \theta \le \pi$. Hence, show that the roots of the equation $8x^3 - 6x - 1 = 0$ are $\cos \frac{\pi}{9}$, $\cos \frac{5\pi}{9}$ and $\cos \frac{7\pi}{9}$.

Hence, find the exact value of $\sec \frac{\pi}{9} + \sec \frac{5\pi}{9} + \sec \frac{7\pi}{9}$.