Dividing Polynomials — Definition, Formula & Examples

Dividing polynomials is the process of finding how many times one polynomial (the divisor) fits into another polynomial (the dividend), producing a quotient and possibly a remainder.

Given polynomials $f(x)$ and $d(x)$ with $d(x) \neq 0$ , polynomial division finds unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that $f(x) = d(x) \cdot q(x) + r(x)$ , where the degree of $r(x)$ is less than the degree of $d(x)$ , or $r(x) = 0$ .

Key Formula

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

Where:

$f(x)$ = The dividend (polynomial being divided)
$d(x)$ = The divisor (polynomial you divide by)
$q(x)$ = The quotient (result of the division)
$r(x)$ = The remainder (what is left over)

How It Works

The two main methods are polynomial long division and synthetic division. Long division works like numerical long division: divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by that result, subtract, and repeat with the new polynomial. Synthetic division is a shortcut that works only when dividing by a linear binomial of the form

(x - c)

. In either method, if a power of

x

is missing from the dividend, you must insert a placeholder term with a coefficient of zero. The process ends when the remaining polynomial has a lower degree than the divisor.

Worked Example

Problem: Divide

(2x^3 + 3x^2 - 5x + 6)

(x + 3)

Step 1: Divide the leading term of the dividend by the leading term of the divisor.

\frac{2x^3}{x} = 2x^2

Step 2: Multiply

(x + 3)

2x^2

and subtract from the dividend.

(2x^3 + 3x^2) - (2x^3 + 6x^2) = -3x^2

Step 3: Bring down

-5x

. Divide

-3x^2

x

to get

-3x

. Multiply and subtract.

(-3x^2 - 5x) - (-3x^2 - 9x) = 4x

Step 4: Bring down

+6

. Divide

4x

x

to get

4

. Multiply and subtract.

(4x + 6) - (4x + 12) = -6

Answer: The quotient is

2x^2 - 3x + 4

with a remainder of

-6

. So

\frac{2x^3 + 3x^2 - 5x + 6}{x + 3} = 2x^2 - 3x + 4 - \frac{6}{x+3}

Why It Matters

Dividing polynomials is essential for factoring higher-degree expressions and applying the Remainder Theorem and Factor Theorem. You will rely on it heavily in Algebra 2 and Precalculus when finding zeros of polynomial functions and performing partial fraction decomposition in calculus.

Common Mistakes

Mistake: Forgetting to include a

0

placeholder for missing powers of

x

in the dividend.

Correction: Before dividing, write every power of

x

from the highest degree down to the constant. For example, write

x^3 + 4

x^3 + 0x^2 + 0x + 4

so that terms align correctly during subtraction.