Polynomial division finds a quotient and a remainder whose combination reproduces the dividend. Over real coefficients, the divisor must be a nonzero polynomial. The process stops when the remainder is zero or its degree is lower than the divisor’s degree, not the quotient’s degree.
Arrange powers before starting long division
Write terms in descending powers and leave a zero coefficient where a power is missing. Divide the leading term of the current remainder by the divisor’s leading term, multiply the entire divisor by that result and subtract the whole product. Align equal powers so signs and coefficients stay together.
Dividing by a monomial can still leave a remainder
For the quadratic below divided by x, the polynomial quotient is 3x + 2 and the remainder is 2. As a rational expression, the division includes the extra fraction and requires x to be nonzero.
Discarding that final 2 would make the result incomplete. If the dividend has lower degree than the divisor, division remains possible: the quotient is zero and the remainder is the original dividend.
Worked long division with a quadratic divisor
Take the cubic A and quadratic B below. Their leading terms give the first quotient term, 3x. Multiply every term of B by 3x and subtract from A; the remaining quadratic has a negative leading coefficient.
Second step: the next quotient term is −1
Divide the new leading term by the leading term of B. Multiply B by −1, then subtract that product. In particular, subtracting its negative coefficients changes their signs. The remaining linear polynomial now has degree less than two, so stop.
Verify the division instead of dropping the remainder
Multiply B by the quotient 3x − 1 and add the remainder −3x + 1. Expanding and combining like powers recovers A exactly. The polynomial identity holds for every x. Only when expressing A divided by B as a rational function must you exclude values where B is zero.
Synthetic division is a shortcut for suitable linear divisors; it should not replace the general method without checking its assumptions. For careful expansion of powers, see the binomial theorem.
