A polynomial in x is a sum of terms of the form axⁿ, where each n is a whole number (0, 1, 2, …). Its degree is the highest power of x, the leading coefficient is the coefficient of that term, and the constant term is the term without x. A polynomial is monic if its leading coefficient is 1.

Worked example. P(x) = 4x³ − 2x⁵ + x − 7 has degree 5, leading coefficient −2 and constant term −7; it is not monic. P(1) = 4 − 2 + 1 − 7 = −4.

Dividing P(x) by D(x) gives a quotient Q(x) and a remainder R(x) with P(x) = D(x)Q(x) + R(x), where the degree of R(x) is less than the degree of D(x).

Long division: divide the leading term of what is left by the leading term of the divisor, multiply the divisor by that term, subtract, and repeat. Write in a 0 for any missing power of x.

Worked example. (x³ + 2x² − 5x + 6) ÷ (x − 1): x² gives x³ − x², leaving 3x² − 5x + 6; 3x gives 3x² − 3x, leaving −2x + 6; −2 gives −2x + 2, leaving 4. So Q(x) = x[sup 2] + 3x − 2 and R = 4.

Remainder theorem: when P(x) is divided by x − a, the remainder is P(a). (Put x = a in P(x) = (x − a)Q(x) + R: the first term becomes 0.)

Factor theorem: x − a is a factor of P(x) exactly when P(a) = 0. To divide by x + 2, use a = −2.

Worked example. The remainder when P(x) = x³ − 3x² + 4x − 2 is divided by x − 3 is P(3) = 27 − 27 + 12 − 2 = 10. Since P(1) = 1 − 3 + 4 − 2 = 0, x − 1 is a factor.