Do not use a calculator in answering this question. Given that $x=1+3\mathbf{i}$ is a root of

Question

Do not use a calculator in answering this question. Given that $x=1+3\mathbf{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0$, find the values of the real numbers $a$ and $b$, and the other roots.[5] The complex numbers $z$ and $w$ satisfy the following equations. $w^* + z = 4 - 6\mathbf{i}$ $w - 2z = 1 + 10\mathbf{i}$ Find $z$ and $w$, giving your answers in the form $c + \mathbf{i}d$, where $c$ and $d$ are real numbers.[5]

Accepted Answer

(a) $a=-6$, $b=-40$ and the other roots are $1-3\mathbf{i}$ and $4$. (b) $w=3+2\,\mathbf{i}$ and $z=1-4\,\mathbf{i}$

2024 RI Promo Q7

2024 RI Promo Q7