Complex Numbers

Synthetic Division

Remainder Theorem

If a polynomial function $f(x)$ is divided by $(ax - b)$ then the remainder is $f(\frac{b}{a})$

Factor Theorem

If $f(\frac{b}{a})$, then $(ax - b)$ is a factor of $f(x)$

Sum and Product of Roots Quadratic

If $p$ and $q$ are roots of a quadratic equation $ax^2 + bx + c$ then

$p + q = -\frac{b}{a}$ and $pq = \frac{c}{a}$

Sum and Products of Roots Polynomials

$a_nx^n+ a_{a-1}x^{n-1}+...+a_1x+a_0 = 0$

the sum of the roots is $-\frac{a_{a-1}}{a_n}$

the products of roots is $(-1)^n\frac{a_0}{a_n}$

Cartesian Form

Form

$z = a + bi$

Conjugate

$z^* = a - bi$

$zz^* = a^2 + b^2$

This is really important to recognize when you see the addition of two squares

Modulus Agreement Form

Modulus

modulus = $|z| = \sqrt{a^2+b^2}$

Agreement

agreement = $\theta = arctan(\frac{b}{a})$

Polar Form

$z = r(\cos\theta + i\sin\theta) = r * cis(\theta)$

Euler Form

$cis(\theta) = e^{i\theta}$

Complex Roots

You can solve for complex roots when your determinate of a quadratic is negative answer

You can also solve knowing that $a^2 + b^2 = (a + bi)(a - bi)$

De Moivre's Theorem

De Moivre's: $(r(cis(\theta)))^n = r^n(cis(n\theta))$

You can solve the same thing using the binomial expansion of $(\cos\theta + i \sin\theta)^n$

Complex Roots of Number

The solutions of $z^n = w$ form a regular n-gon with vertices on a circle of radius $|z|$ centered at the origin

Roots of Unity

the roots of unity are the solutions of $z^n = 1$

the roots are $1, cis(\frac{2\pi}{n}), cis(\frac{4\pi}{n}), ... cis(\frac{2(n - 1)\pi}{n})$