Equations of higher degree

Complex numbers, grade 12 (5 units) Lesson 7 of 18Learning

Equations of higher degree

Conjugate solutions

In an equation $\displaystyle {p(z)}={0}$ , with $\displaystyle p$ a polynomial with real coefficients: if $\displaystyle z_0$ is a solution, so is $\displaystyle \bar{z}_0$ . The proof below uses the properties of the conjugate:

The conjugate of a sum is the sum of the conjugates, and the conjugate of a product is the product of the conjugates; so $\displaystyle {\overline{z^n}}={\bar{z}^n}$ .
A real coefficient $\displaystyle a$ equals its conjugate, so $\displaystyle {\overline{a z^n}}={a\bar{z}^n}$ .
So $\displaystyle {\overline{p(z_0)}}={p(\bar{z}_0)}$ , and $\displaystyle {p(z_0)}={0}$ gives $\displaystyle {p(\bar{z}_0)}={\bar{0}}={0}$ .

So an equation of the third degree with real coefficients always has at least one real solution: the solutions that are not real come in pairs.

The biquadratic equation

In an equation that holds only $\displaystyle z^4$ , $\displaystyle z^2$ and a number, substitute $\displaystyle {t}={z^2}$ . The example below solves $\displaystyle {z^4+3z^2-4}={0}$ :

$\displaystyle {t^2+3t-4}={0}$ , so $\displaystyle {t}={1}$ or $\displaystyle {t}={-4}$ .
$\displaystyle {z^2}={1}$ gives $\displaystyle {z}={\pm1}$ ; $\displaystyle {z^2}={-4}$ gives $\displaystyle {z}={\pm2i}$ .
Four solutions: $\displaystyle 1$ , $\displaystyle -1$ , $\displaystyle 2i$ , $\displaystyle -2i$ .

Worked example: guessing and dividing

When you know one solution $\displaystyle z_0$ , the polynomial divides by $\displaystyle z-z_0$ , and an equation of lower degree is left. The example below solves $\displaystyle {z^3-3z^2+4z-2}={0}$ :

A whole-number solution divides the constant term $\displaystyle -2$ . Trying: $\displaystyle {1-3+4-2}={0}$ , so $\displaystyle {z}={1}$ is a solution.
Dividing by $\displaystyle z-1$ : $\displaystyle {z^3-3z^2+4z-2}={(z-1)(z^2-2z+2)}$ .
$\displaystyle {z^2-2z+2}={0}$ : $\displaystyle {\Delta}={-4}$ , so $\displaystyle {z}={1\pm i}$ .
The solutions: $\displaystyle 1$ , $\displaystyle 1+i$ , $\displaystyle 1-i$ .

When a solution $\displaystyle z_0$ that is not real is given in an equation with real coefficients, divide by $\displaystyle (z-z_0)(z-\bar{z}_0)$ – a polynomial of the second degree with real coefficients.

Exercise 1

Solve with the substitution $\displaystyle {t}={z^2}$ .

Exercise 2

Solve: factor, or find one solution and divide.

Exercise 3

$\displaystyle 2+i$ is a solution of the equation $\displaystyle {z^3-5z^2+9z-5}={0}$ .
Find the other solutions.

Exercise 4

Build an equation of the third degree with real coefficients, two of whose solutions are $\displaystyle 1+i$ and $\displaystyle -1$ .
What is the third solution? Write the equation, and check it in the equation tool.

Exercise 5

The equation $\displaystyle {z^4+az^3+bz^2+cz+d}={0}$ has real coefficients, and $\displaystyle i$ and $\displaystyle 2-3i$ are solutions of it.
Find all its solutions, and explain why it has no real solution.

Formulas: complex numbers, grade 12 (5 units)

Open as a page

The algebraic form and the four operations

The number $\displaystyle i$
$\displaystyle i^{2}=-1$
The algebraic form
$\displaystyle z=a+bi$ $\displaystyle a=\operatorname{Re}(z),\ b=\operatorname{Im}(z)$
Equality of complex numbers
$\displaystyle a+bi=c+di\iff a=c,\ b=d$
Multiplying
$\displaystyle (a+bi)(c+di)$ $\displaystyle =$ $\displaystyle (ac-bd)+(ad+bc)i$
The conjugate
$\displaystyle \bar{z}=a-bi$
A number times its conjugate
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}$
Dividing – times the denominator’s conjugate
$\displaystyle \frac{z_{1}}{z_{2}}$ $\displaystyle =$ $\displaystyle \frac{z_{1}\cdot\bar{z}_{2}}{z_{2}\cdot\bar{z}_{2}}$

The modulus and the conjugate

The modulus – the distance from the origin
$\displaystyle |z|$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The distance between two points
$\displaystyle |z_{1}-z_{2}|$ $\displaystyle =$ $\displaystyle \sqrt{(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}}$
The product with the conjugate and the modulus
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle |z|^{2}$
The modulus of a product
$\displaystyle |z_{1}\cdot z_{2}|$ $\displaystyle =$ $\displaystyle |z_{1}|\cdot|z_{2}|$
The conjugate of a sum
$\displaystyle \overline{z_{1}+z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}+\bar{z}_{2}$
The conjugate of a product
$\displaystyle \overline{z_{1}\cdot z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}\cdot\bar{z}_{2}$
A number plus its conjugate
$\displaystyle z+\bar{z}$ $\displaystyle =$ $\displaystyle 2\operatorname{Re}(z)$
The triangle inequality
$\displaystyle |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|$

The quadratic equation

The quadratic formula
$\displaystyle z_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
The sum of the roots
$\displaystyle z_{1}+z_{2}$ $\displaystyle =$ $\displaystyle -\frac{b}{a}$
The product of the roots
$\displaystyle z_{1}\cdot z_{2}$ $\displaystyle =$ $\displaystyle \frac{c}{a}$

The polar form (angles in radians)

The polar form
$\displaystyle z=r(\cos\theta+i\sin\theta)$ $\displaystyle r=|z|$
The notation cis
$\displaystyle r\operatorname{cis}\theta$ $\displaystyle =$ $\displaystyle r(\cos\theta+i\sin\theta)$
The real part
$\displaystyle a=r\cos\theta$
The imaginary part
$\displaystyle b=r\sin\theta$
The angle – by the quadrant of the point
$\displaystyle \tan\theta=\frac{b}{a}$ $\displaystyle a\neq0$
A product in polar form
$\displaystyle z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\left[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\right]$
A quotient in polar form
$\displaystyle \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\operatorname{cis}(\theta_{1}-\theta_{2})$
De Moivre’s theorem
$\displaystyle \left[R(\cos\varphi+i\sin\varphi)\right]^{n}$ $\displaystyle =$ $\displaystyle R^{n}(\cos n\varphi+i\sin n\varphi)$
The solutions of $\displaystyle {z^n}={R(\cos\varphi+i\sin\varphi)}$
$\displaystyle z_{k}=\sqrt[n]{R}\left[\cos\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)\right]$ $\displaystyle z^{n}=R(\cos\varphi+i\sin\varphi),\ k=0,1,\ldots,n-1$

Loci and rotation

The circle with centre $\displaystyle z_0$ and radius $\displaystyle R$
$\displaystyle |z-z_{0}|=R$
The perpendicular bisector of the segment $\displaystyle z_1z_2$
$\displaystyle |z-z_{1}|=|z-z_{2}|$
Apollonius’ circle
$\displaystyle |z-z_{1}|=m|z-z_{2}|$ $\displaystyle m>0,\ m\neq1$
Rotating by $\displaystyle \theta$ about $\displaystyle z_0$
$\displaystyle w=z_{0}+(z-z_{0})\operatorname{cis}\theta$