Polynomial. Division of polynomial by polynomial

Definition: a Polynomial of one variable is a polynomial of the form where the numerical coefficients.

Definition: If this polynomial is called the polynomial of the first degree relatively variable .

Member called a senior member of the polynomial a — it free member.

— the polynomial of the third degree.

Identically equal polynomials in one variable

Definition: Two polynomials are called equalif they take equal values for all values of the variable.

Properties identical with equality of polynomials in one variable

  1. If the polynomial is identically equal to zero (i.e. has zero values at all values ), then all of its coefficients are equal to zero.
  2. If two polynomial identically equal to (i.e., acquire the same value at all values ), then they coincide (i.e., their degrees are equal and the coefficients of equal powers of equal).

Division of polynomial by polynomial

Definition: If two polynomials it is possible to find a polynomial , that is divided into .

Example

Since , the polynomial is divisible by the polynomial

The division of the polynomial by the polynomial s Stacey

Definition: a Polynomial is divided by the polynomial s Stacey, if you can find a pair of polynomials , and the degree of the remainder of smaller degree .

If the remainder , then the polynomial is divisible by the polynomial without a remainder)

Example

,

The division of the polynomial by the polynomial "area"

The rule of division of polynomials in one variable

  1. To place the members of the polynomials with descending exponents of the variable.
  2. To share a senior member of the dividend on the senior member of the divider.
  3. The result is multiplied by the divisor and subtract this product from the dividend.
  4. With the obtained difference perform the same operation: divide it senior member the senior member of the divider and the result again multiplied by the divisor, and so on. This process continue to give until I get the balance to zero (if one polynomial divided by another) or as long as the balance does not get the polynomial degree is less than degree of divisor.

Theorem Continuous

The remainder of dividing the polynomial on doclen equal

Corollary: If is a root of the polynomial (i.e., ), then this polynomial is divided .

Example

The remainder of dividing the polynomial on doclen equal , that is divided into without a remainder.

Dividing into "area" or Horner's scheme, we get:

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