Polynomial long division works like numerical long division: divide the leading terms, multiply back, subtract and bring down the next term. It is used to simplify rational functions, find oblique asymptotes and factor polynomials once a root is known.
How to use the Polynomial Division Calculator
- Choose the degree of each polynomial – the matching coefficient fields appear.
- Enter the coefficients from the highest power down; leave a field at 0 for a missing power.
- Read the result, written with powers of x in descending order.
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How it works
The leading coefficient of the remainder is repeatedly divided by the leading coefficient of Q(x) until the remainder’s degree is less than Q’s degree. A zero remainder means Q(x) is a factor of P(x).
Formula
P(x) = Q(x) · S(x) + R(x), deg R < deg Q
Worked example
Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): -6; a₁ (x): 11; a₀: -6; Degree of Q(x): 1; b₁ (x): 1; b₀: -1.
Results — Quotient S(x) = x² − 5x + 6; Remainder R(x) = 0; Dividend P(x) = x³ − 6x² + 11x − 6; Divisor Q(x) = x − 1; Check = P(x) = (x − 1)(x² − 5x + 6).
Key terms
- Quotient
- The polynomial S(x) in P = QS + R.
- Remainder
- The polynomial R(x) left over, of lower degree than Q.
Frequently asked questions
What does a zero remainder mean?
The divisor is a factor of the dividend.
When can I use synthetic division instead?
When the divisor is linear, x − r.
What is the degree of the quotient?
deg P − deg Q.
How do I write the answer?
P/Q = S + R/Q.