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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 proof.
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
_{0}
b
^{m-1}
+ a
_{0}
b
^{m-2}
+ …+ a
_{m}
= N .
The remark.
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.
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