# Division of polynomial by linear binomial

Linear binomial. Bezout's theorem.

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 a 0 x m + a 1 x m -

1 + a 2 x m - 2 + …+ a m is divided by   x – b with a remainder   N = a 0 b m + a 1 b m - 1 + a 2 b m - 2 + …+ a m .

The  p r o o f . According to the definition of division (see above) we have:

a 0 x m + a 1 x m -

1 + a 2 x m - 2 + …+ a m = ( x – b ) Q + N ,

where Q is some polynomial, N is some number. Substitute here x = b , then ( x – b ) Q will be missing and we receive:

a 0 b m + a 1 b m -
1 + a 2 b m - 2 + …+ a m = N .

The  r e m a r k . It is possible, that N = 0 . Then b is a root of the equation:

a 0 x m + a 1 x m -

1 + a 2 x m - 2 + …+ a m = 0 .

The theorem has been proved.