Division of polynomial by linear binomial
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:
where Q is some polynomial, N is some number. Substitute here x = b , then ( x – b ) Q will be missing and we receive:
The r e m a r k . It is possible, that
N = 0
is a root of the equation:
The theorem has been proved.