Polynomial Division

Divide by a monomial, then long-divide one polynomial by another.

The idea

Polynomial division rewrites a quotient of polynomials as a polynomial plus a remainder, just as $\tfrac{17}{5}$ is $3$ with remainder $2$: the $\dfrac{\text{dividend}}{\text{divisor}}$ becomes a polynomial quotient plus a remainder, and the remainder is $0$ if and only if the divisor divides evenly.

When the divisor is a single term, split the fraction over the terms of the numerator and divide term by term, dividing the coefficients and subtracting the exponents in each piece.

When the divisor has more than one term, use long division, one round at a time.

Algorithm.

Algorithm: Polynomial Long Division Input: a dividend and a divisor with more than one term Output: a quotient Q and a remainder R with deg R < deg divisor 1. R = dividend, Q = 0 // invariant: dividend = divisor × Q + R 2. if deg R < deg divisor, return Q and R 3. t = (leading term of R) / (leading term of divisor) 4. Q = Q + t, R = R - t × divisor // leading terms cancel, so deg R drops 5. go to step 2

Ways to work on it

• Walkthrough. Monomial division, then full polynomial long division.
• Practice. Divide a quadratic by a monomial.
• Hardest. Long division that leaves a remainder.

Not sure where to start? Take the ten-question placement test.