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 a0 xm + a1 xm-1
+ a2 xm-2 + …+ am is divided by x – b with a remainder N = a0 bm + a1 bm-1 + a2 bm-2 + …+ am .
The p r o o f . According to the definition of division (see above) we have:
a0 xm + a1 xm-1
+ a2 xm-2 + …+ am = ( 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:
a0 bm
+ a0 bm-1 + a0 bm-2 + …+ am = N .
The r e m a r k . It is possible, that N = 0 . Then b is a root of the equation:
a0 xm + a1 xm-1
+ a2 xm-2 + …+ am = 0 .
The theorem has been proved.
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