Imaginary

$i=\sqrt{-1}$. All complex numbers have a real part and an imaginary part. The modulus is $\sqrt{Re+Im}$ When its at the bottom of a fraction, you can multiply by the conjugate to make it more normal. The method leverages the difference of squares.

Division with Polynomials

Synthetic division

Coefficients inside the box. The root in the outside. Sum down a column, multiply difference by the root and bring to next box. The remaining solution is one degree less in every term in comparison to prior.

Long Division

Line the digits up, but instead of place values, its the degree of the terms that sets the places.

Remainder Theorem

When polynomial $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$.

Factor theorem

Builds off of remainder theorem. If you substitute in the $f(a)$ to obtain zero, then $(x-a)$ is a factor.

Complex roots

If a real polynomial has complex roots, the complex roots must come in conjugate pairs. Quite a helpful fact. All synthetic division stuff work with this too.

Sum and Product rule

For a polynomial ${\sum }_{r=0}^n{a}_r{x}^r$ Sum of roots is $-\frac{a_{n-1}}{a_n}$ and the product is $\frac{(-1{)}^n{a}_0}{a_n}$ Summary: Sum is second term divided by first term. Product is last term divided by first term, alternating signs every time the degree increases.

Solving systems

Matrix of coefficents and then the result on the rightmost column. Rref() it to get the solution.