Math topic中文

Conjugate roots polynomial

Conjugate roots of a polynomial equation with real coefficients.

Practice questions

1Given that $z = 2 - \text{i}$ is a root of the equation $4z^4 - 12z^3 + 17z^2 + pz + q = 0$ , where $p$ and $q$ are real,…
2One of the roots of the equation $2z^4 - 14z^3 + 33z^2 - 26z + p = 0\text{, where } p \text{ is a constant}$ is $3 + \mathrm{i}$ …
3Do not use a calculator in answering this question. Given that $x = 1 + 3\mathrm{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0,$ …
4One of the roots of the equation $2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0\text{, where } p \text{ is a constant}$ is…
5The function $\text{f}$ is defined by $\text{f}(z) = z^4 + Az^3 + Bz^2 + Cz + 45$ , where $A$ , $B$ and $C$ are real numbe…
6The equation $z^{3}-2z+k=0$ , where $k$ is a real constant, has a root $z = 1 + ai$ , where $a$ is a positive real constan…
7A complex number $z$ varies with $t$ such that $z = 2\cos t + \mathrm{i}(3\sin t), \quad \text{where } 0 \le t < 2\pi.$ …
8Do not use a graphing calculator in answering this question. The complex number $w$ is a root to the equation $z^4 + a z^3 + 46 z^2 + b z + 125 = 0 ,$ …
9One of the roots of the equation $2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0,$ where $p$ is a constant, is $3 + i$ . Base…
10The cubic equation $x^3 + ax^2 + bx + c = 0$ , where $a$ , $b$ and $c$ are constants, has roots $3 + \mathrm{i}$ and $2$ .…
11It is given that $\mathrm{f}(x) = 2x^6 - 4x^4 - 6x^2 - 7$ . The diagram shows the curve with equation $y = \mathrm{f}(x)$ …
12The complex number $z$ is such that $az^2 + bz + a = 0$ where $a$ and $b$ are real constants. It is given that $z = z_0$ …
13The three distinct roots of the equation $x^3 - 1 = 0$ are denoted by $1, \omega$ and $\omega^2$ . Without first finding…
14Do not use a calculator in answering this question. One of the roots of the equation $3x^3 + px^2 + qx + 3 = 0$ , whe…
15One of the roots of the equation $z^4-4z^3+az^2+bz+78=0$ , where $a$ and $b$ are real, is $3+2i$ . Find the values of $a$ …
16Find the complex number z in the form x+iy, where x, y \in \mathbb{R} such that $\dfrac{3i z}{z^{*}-1+2i} = -i$ . [Note:…
17Two of the roots of the equation $z^3 + az^2 + bz + c = 0$ are $3\text{e}^{\text{i}\left(-\frac{2}{3}\pi\right)}$ and $-2$ …
18The polynomial $\text{P}(z)$ is given by $z^3 - 8z + a$ , where $a$ is a real constant. Given that $\sqrt{3} + \text{i}$ …
19[Cambridge/9709/2009] The complex number $-2 + i$ is denoted by $u$ . Given that $u$ is a root of the equation $x^3 - 11x - k = 0$ …
20The function f is defined by $f(x)=ax^3+bx^2+cx+d$ , where $a,b,c$ and $d$ are real numbers. Given that $1-2i$ and $-1$ a…
21Do not use a calculator in answering this question. The equation $2z^3 + 5z^2 + pz + q = 0$ has a root $z = -1 + \mathrm{i}$ …
22One of the roots of the equation $z^4 - z^3 - 9z^2 + sz + t = 0$ , where $s$ and $t$ are real, is $1 + 2\text{i}$ . Find t…