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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  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 )will be missing and we receive:

a0 bm + a1 bm-1 + a2 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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