Polynomial Division

Long Division

Same process as numerical long division!

Steps:

1. Divide leading terms
2. Multiply divisor by quotient term
3. Subtract
4. Bring down next term
5. Repeat

Example: $(x^2 + 5x + 6) \div (x + 2)$

$\begin{array}{c|cc} & x + 3 \\ \hline x + 2 & x^2 + 5x + 6 \\ & x^2 + 2x \\ \hline & 3x + 6 \\ & 3x + 6 \\ \hline & 0 \end{array}$

Result: $x + 3$ with remainder $0$

Synthetic Division

Only works when dividing by $(x - c)$

Much faster than long division!

Steps:

1. Write coefficients of dividend
2. Use $c$ from divisor $(x - c)$
3. Bring down first coefficient
4. Multiply and add repeatedly

Example: $(2x^3 - 5x^2 + 3x - 2) \div (x - 2)$

Use $c = 2$: $\begin{array}{c|cccc} 2 & 2 & -5 & 3 & -2 \\ & & 4 & -2 & 2 \\ \hline & 2 & -1 & 1 & 0 \end{array}$

Result: $2x^2 - x + 1$ with remainder $0$

Remainder Theorem

When dividing $f(x)$ by $(x - c)$: $\text{Remainder} = f(c)$

Division Algorithm

$f(x) = q(x) \cdot d(x) + r(x)$

Where $q$ is quotient, $d$ is divisor, $r$ is remainder