Bezou theorem

Bezout 's theorem states that the remainder of the division of a polynomial $P(x)$ ${\ displaystyle P (x)}$ $P (x)$ on the binomial $(x-a)$ ${\ displaystyle (xa)}$ $(x-a)$ is equal to $P(a)$ ${\ displaystyle P (a)}$ $P (a)$ .

It is assumed that the coefficients of the polynomial are contained in some commutative ring with unity (for example, in the field of real or complex numbers ).

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Proof

Divide the polynomial with the remainder $P(x)$ ${\ displaystyle P (x)}$ on the binomial $x-a$ ${\ displaystyle xa}$ :

P(x)=(x-a)Q(x)+R(x),

{\ displaystyle P (x) = (xa) Q (x) + R (x),}

Where $R(x)$ ${\ displaystyle R (x)}$ - the remainder. Because $\deg R(x)<\deg(x-a)=1$ ${\ displaystyle \ deg R (x) <\ deg (xa) = 1}$ then $R(x)$ ${\ displaystyle R (x)}$ - a polynomial of degree not higher than 0, that is, a constant. Substituting $x=a$ ${\ displaystyle x = a}$ , insofar as $(a-a)Q(a)=0$ ${\ displaystyle (aa) Q (a) = 0}$ , we have $P(a)=R(a)$ ${\ displaystyle P (a) = R (a)}$ .

Consequences

Number $a$ ${\ displaystyle a}$ is the root of the polynomial $p(x)$ ${\ displaystyle p (x)}$ if and only if $p(x)$ ${\ displaystyle p (x)}$ divides without remainder into a binomial $x-a$ ${\ displaystyle xa}$ (From this, in particular, it follows that the set of roots of the polynomial $P(x)$ ${\ displaystyle P (x)}$ identical to the set of roots of the corresponding equation $P(x)=0$ ${\ displaystyle P (x) = 0}$ )
The free term of the polynomial is divided by any integer root of the polynomial with integer coefficients (if the leading coefficient is 1, then all rational roots are integer).
Let be $a$ ${\ displaystyle a}$ Is the whole root of the reduced polynomial $A(x)$ ${\ displaystyle A (x)}$ with integer coefficients. Then for any whole $k$ ${\ displaystyle k}$ number $A(k)$ ${\ displaystyle A (k)}$ divided by $a-k$ ${\ displaystyle ak}$ .

Applications

Bezout's theorem and its corollaries make it easy to find rational roots of polynomial equations with rational coefficients.

Bezou theorem

Content

Proof

Consequences

Applications

See also

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