Polynomial Division
Division leaves a quotient and a remainder
A polynomial is a sum of constant multiples of non-negative integer powers of . For , the degree is 3, the highest power; the leading term is , and its coefficient is 2. Expressions containing or are not polynomials in .
Integer division gives : divide 17 by 5 to get quotient 3, remainder 2. Polynomial division has the same structure:
is the dividend (what is divided); the divisor; the quotient; what remains. Division is finished when the remainder is zero or has lower degree than the divisor.
This is an identity: true for every . A fraction form additionally requires .
Cancel the highest remaining power
Worked example
Divide x³ − 4x + 1 by x − 2.
Write . The zero coefficient keeps the missing power in its column.
Divide leading terms: . Multiply the whole divisor by , then subtract that product, including both signs.
Repeat: . Subtract :
Stop: a constant has lower degree than the linear divisor. The quotient collects the terms we used, ; the remainder is 1.
Expand the right side to check all coefficients. In particular, restores the missing term.
Divide by . State quotient and remainder and check by multiplication.
Show worked answer
The successive quotient terms are , , then −3.
Indeed . A polynomial remainder may be negative.