Dividing Polynomials: Long Division and Synthetic Division

By the end of this lesson, you will be able to divide polynomials using long division, apply the faster synthetic division for linear divisors, interpret the quotient and remainder, and use division to verify factors.

The Big Idea

Polynomial division works exactly like numeric long division, and it solves a critical problem: given that $r$ is a zero of a polynomial, dividing by $(x - r)$ factors out that zero and reduces the degree by one. Synthetic division streamlines this for linear divisors, making what would be tedious into a simple number table.

Polynomial Long Division

Setup: $\dfrac{p(x)}{d(x)} = q(x) + \dfrac{r(x)}{d(x)}$ where $\deg(r) < \deg(d)$.

Algorithm:

1. Divide the leading term of the dividend by the leading term of the divisor
2. Multiply the result by the entire divisor
3. Subtract from the dividend
4. Bring down the next term
5. Repeat until the remainder has degree less than the divisor

Example 1 — Long Division

Divide $2x^3 - 3x^2 + x - 5$ by $x - 2$.

$$\begin{array}{r} 2x^2 + x + 3 \ x - 2 \overline{)2x^3 - 3x^2 + x - 5} \ -(2x^3 - 4x^2) \ \hline x^2 + x \ -(x^2 - 2x) \ \hline 3x - 5 \ -(3x - 6) \ \hline 1 \end{array}$$

Result: $2x^2 + x + 3$ with remainder $1$.

$$\frac{2x^3 - 3x^2 + x - 5}{x - 2} = 2x^2 + x + 3 + \frac{1}{x-2}$$

Synthetic Division

2 | 2   -3    1   -5
   |      4    2    6
   |-----------------
     2    1    3    1

Works only when the divisor is linear: $x - r$ (or $x + r$, where $r$ is negative).

Setup: write just the coefficients of the dividend; use $r$ (not $x - r$) in the box.

Example: Divide $2x^3 - 3x^2 + x - 5$ by $x - 2$ ($r = 2$):

Read: quotient coefficients $2, 1, 3$ → $2x^2 + x + 3$; remainder $= 1$.

Same answer as long division. ✓

Example 2 — Synthetic Division with Remainder 0

Divide $x^3 - 7x + 6$ by $x - 2$.

Missing $x^2$ term → use coefficient 0.

 2 | 1    0   -7    6
   |      2    4   -6
   |------------------
     1    2   -3    0

Quotient: $x^2 + 2x - 3$; remainder: $0$.