Linear binomial is a polynomial of the first degree: ax+ b. If to divide a polynomial, containing a letter x, by a linear binomial x – b, where b is a number ( positive or negative ), then a remainder will be a polynomial only of zero degree, i.e. some number N , which can be found without finding a quotient. Exactly, this number is equal to the value of the polynomial, received at x = b. This property is proved by Bezout’s theorem: a polynomial a₀ x^m + a₁ x^m-1 + a₂ x^m-2 + …+ a_m is divided by x – b with a remainder N = a₀ b^m + a₁ b^m-1 + a₂ b^m-2 + …+ a_m .

The proof. According to the definition of division (see above) we have: where Q is some polynomial, N is some number. Substitute here x = b , then ( x– b ) Q will be missing and we receive: The remark. It is possible, that N = 0 . Then b is a root of the equation: The theorem has been proved.