Polynomial Quotient — Definition, Formula & Examples

A polynomial quotient is the result obtained when one polynomial is divided by another polynomial, not counting any remainder.

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

How It Works

To find a polynomial quotient, you divide the dividend polynomial by the divisor polynomial using either long division or synthetic division. At each step, you divide the leading term of the current dividend by the leading term of the divisor, write that result as the next term of the quotient, then subtract and bring down terms. The process continues until the remaining polynomial has a degree lower than the divisor. That remaining polynomial is the remainder, while the accumulated terms form the quotient.

Worked Example

Problem: Find the quotient when f(x) = 2x³ + 5x² − x + 6 is divided by d(x) = 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 2x² by (x + 3) and subtract from the dividend to get the new partial dividend.

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

Step 3: Repeat: divide −x² by x to get −x, multiply and subtract.

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

Step 4: Divide 2x by x to get 2, multiply and subtract.

2x + 6 - (2x + 6) = 0

Answer: The polynomial quotient is q(x) = 2x² − x + 2, with remainder 0.

Why It Matters

Finding polynomial quotients is essential for factoring higher-degree polynomials and locating their roots. In precalculus and calculus, you use polynomial division to simplify rational expressions, perform partial fraction decomposition, and identify oblique asymptotes of rational functions.

Common Mistakes

Mistake: Forgetting to include a 0 placeholder for missing degree terms in the dividend.

Correction: If a power of x is missing (e.g., no x² term), insert 0x² before dividing so that alignment stays correct throughout the long division.