Algebra

Polynomial Long Division Calculator | Step-by-Step Quotient & Remainder

Divide any polynomial P(x) by any divisor Q(x) using full long-division notation. Shows every multiply-subtract step exactly as you would write it by hand, outputs the quotient and remainder, reconstructs the original polynomial for verification, and checks if Q(x) is an exact factor.

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Enter coefficients in descending order, space-separated. Example: 1 -2 1 -3 = x³−2x²+x−3

Dividend P(x) — coefficients

Divisor Q(x) — coefficients

What Is the Polynomial Long Division Calculator | Step-by-Step Quotient & Remainder?

Polynomial long division is the analogue of integer long division for polynomials. At each step, you divide the leading term of the current remainder by the leading term of the divisor to get the next quotient term, multiply the divisor by that term, subtract, and repeat until the remainder has lower degree than the divisor. If the remainder is zero, the divisor is an exact factor — this is the basis of the Factor and Remainder Theorems.

Formula

P(x) ÷ D(x) = Q(x) remainder R(x)

Verification: P(x) = D(x) · Q(x) + R(x)

Exact factor: D(x) | P(x) iff R(x) = 0

Step: divide leading term, multiply D, subtract, repeat

How to Use

1
Enter dividend coefficients from highest to lowest degree (e.g. 1 -6 11 -6 for x³−6x²+11x−6)
2
Enter divisor coefficients the same way (e.g. 1 -1 for x−1)
3
Or click a preset to load an example immediately
4
Click Divide to run the long division
5
Read each step: term added to quotient, multiplication, subtraction
6
Check the verification: P(x) = D(x)·Q(x) + R(x)

Enter the dividend P(x) as space-separated coefficients from highest to lowest degree (e.g., '2 3 -1 5' for 2x³+3x²−x+5). Enter the divisor D(x) the same way. Click Divide to see each step of the long division process, the quotient, and remainder.

Example Calculation

Example: (x³ − 6x² + 11x − 6) ÷ (x − 1)

Step 1: x³ ÷ x = x² → multiply: x²(x−1) = x³−x² → subtract → −5x²+11x−6

Step 2: −5x² ÷ x = −5x → multiply: −5x(x−1) = −5x²+5x → subtract → 6x−6

Step 3: 6x ÷ x = 6 → multiply: 6(x−1) = 6x−6 → subtract → 0

Result: Q(x) = x²−5x+6, R = 0 → (x−1) is a factor ✓

Frequently Asked Questions

What is the Remainder Theorem?

The Remainder Theorem states that when a polynomial P(x) is divided by (x−a), the remainder equals P(a). So if you want to find P(3) for P(x)=x³−6x²+11x−6, divide by (x−3) and the remainder is P(3). If P(3)=0, then x=3 is a root.

What is the Factor Theorem?

The Factor Theorem says (x−a) is a factor of P(x) if and only if P(a)=0, which is the case when the long division has remainder zero. This is used to factor polynomials once rational roots are found via the Rational Root Theorem.

How do I enter a polynomial with missing terms?

Include zero coefficients for missing terms. For x³+1 enter "1 0 0 1". For x⁴−x² enter "1 0 -1 0 0". Every power from the highest degree down must have its coefficient listed, even if it is zero.

What if the divisor has higher degree than the dividend?

If deg(D) > deg(P), the quotient is 0 and the remainder is P(x) itself. This is analogous to 3÷5 = 0 remainder 3 in integer division. The calculator handles this case gracefully.

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