L3 Complex Numbers
Question Bank
One solution of the quadratic equation $z^2 + bz + c = 0$, where $b$ and $c$ are real, is $z = -2 + \sqrt{5}\,i$.
Find the values of $b$ and $c$.
The cubic equation $z^3 + az^2 + bz + c = 0$ has real coefficients $a$, $b$ and $c$.
Two of its solutions are $z = 2$ and $z = 1 + 3i$.
Find the values of $a$, $b$ and $c$.
One solution of the equation $z^3 - z^2 - z + d = 0$, where $d$ is real, is $z = -1 + 2i$.
Find the value of $d$ and the other two solutions.
One solution of the equation
$ 3z^3 - 4z^2 + kz + 20 = 0 $
where $k$ is real, is $z = 1 + 3i$.
Find the value of $k$ and the other two solutions.
One factor of the polynomial $P(z) = z^3 - 2z^2 - 5z + 6$ is $(z + 2)$.
Find the other two linear factors.
One solution of the equation
$ z^3 + az^2 + 10z - 12 = 0 $
where $a$ is real, is $z = 3$.
Find the value of $a$ and the other two solutions, giving them in exact form. You must show algebraic working.
One solution of the equation
$ z^3 + pz^2 + qz + 10 = 0 $
where $p$ and $q$ are real, is $z = -2 + i$.
Find the values of $p$ and $q$, and the other two solutions.
$z = 3 + 2i$ is one of the roots of the polynomial equation $z^3 - kz^2 + mz - n = 0$, where $k, m$ and $n$ are real.
Find the real root in terms of $k$.
One solution of the equation $z^3 - 10z^2 + Az - 34 = 0$ is $z = 4 - i$.
If $A$ is a real number, find the value of $A$ and the other two solutions of the equation.
One solution of the equation
$ 2z^3 - 3z^2 + kz + 3 = 0 $
where $k$ is real, is $z = 1 - \sqrt{2}\,i$.
Find the value of $k$ and the other two solutions.
One solution of the equation $z^3 - 5z^2 + az + b = 0$ is $z = \sqrt{2}\,\text{cis}\dfrac{3\pi}{4}$.
If $a$ and $b$ are real, find the values of $a$ and $b$, and the other two solutions of the equation.
The equation
$ z^4 - 4z^3 + 9z^2 - 16z + 20 = 0 $
has a solution that is purely imaginary.
Find all four solutions of the equation. You must show algebraic working.
Consider the equation
$ z^3 - 3z^2 + kz + 2 - k = 0 $
where $k$ is a real constant.
Find the values of $k$ for which two of the solutions are not real, and for these values write all three solutions in terms of $k$.
If $x^2 + 2x - 3$ is a factor of the polynomial $p(x) = x^4 + Ax^3 - 3x^2 + Bx + 6$, find the values of $A$ and $B$.
Hence, express $p(x)$ as a product of linear factors.
Solve the equation $z^2 - (3 + i)z + 4 + 3i = 0$.
Give your solutions in the form $a + bi$, showing your algebraic working.