Long Division of Polynomials

How polynomial long division works

The process is identical to integer long division — divide, multiply, subtract, bring down — but with polynomials instead of numbers.

Step-by-step: $(2x^4 + 3x^3 - x^2 + 5x - 4) \div (x^2 + 2)$

Step 1 — Divide leading terms: $2x^4 \div x^2 = 2x^2$. Write $2x^2$ in the quotient.

Step 2 — Multiply: $2x^2 \cdot (x^2 + 2) = 2x^4 + 4x^2$.

Step 3 — Subtract from dividend: $(2x^4 + 3x^3 - x^2) - (2x^4 + 4x^2) = 3x^3 - 5x^2$. Bring down $+5x$.

Step 4 — Repeat. Divide: $3x^3 \div x^2 = 3x$. Multiply: $3x(x^2 + 2) = 3x^3 + 6x$. Subtract: $(3x^3 - 5x^2 + 5x) - (3x^3 + 6x) = -5x^2 - x$. Bring down $-4$.

Step 5 — Repeat. Divide: $-5x^2 \div x^2 = -5$. Multiply: $-5(x^2 + 2) = -5x^2 - 10$. Subtract: $(-5x^2 - x - 4) - (-5x^2 - 10) = -x + 6$.

Step 6 — Stop. The remainder $-x + 6$ has degree 1, which is less than the divisor's degree 2.

Result: $\text{Quotient} = 2x^2 + 3x - 5$, $\text{Remainder} = -x + 6$.

$2x^4 + 3x^3 - x^2 + 5x - 4 = (x^2 + 2)(2x^2 + 3x - 5) + (-x + 6)$

Check: Expand $(x^2 + 2)(2x^2 + 3x - 5) + (-x + 6) = 2x^4 + 3x^3 - 5x^2 + 4x^2 + 6x - 10 - x + 6 = 2x^4 + 3x^3 - x^2 + 5x - 4$ ✓

When to use long division vs synthetic

• Long division: works for ANY divisor (including $x^2 + 2$, $2x - 3$, etc.)
• Synthetic division: only works when dividing by $(x - c)$ (linear, leading coefficient 1)

Try it in the visualization

Each step of the long division is shown like arithmetic long division. The quotient builds up term by term, and the remainder decreases in degree at each step.