# 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.