IB Mathematics Analysis and Approaches Topic AHL…

The complex number $z$ is defined by $z = \frac{4-4i}{1+i}$ .

(a) Find an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

(b) Find the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

The polynomial $p(x) = x^4+ax^3+7x+b$ , where $a$ and $b$ are constants, is given. It is given that $p(x)$ is divisible by $x^2+x-1$ .

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

(b) When $a$ and $b$ have these values, find the real roots of the equation $p(x)=0$ .

Given that $z = r(\cos\theta + i\sin\theta)$ , where $r$ is a positive real number and $\theta$ is a real angle (in radians).

(a) Find and simplify an expression for $z^n$ , where $n$ is a positive integer.

(b) Given a real constant $k$ , find and simplify an expression for $\frac{k}{z^n}$ , where $n$ is a positive integer.

Let $u=-1+i$ and $p(x)=x^4+x^3+3x^2+4x+6$ .

(a) Show your working to verify that $u$ is a root of $p(x)=0$ , and write down a second complex root of the equation.

(b) Find the other two roots of the equation $p(x)=0$ .

IB Mathematics Analysis and Approaches AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank