**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

- 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.
- 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

### 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

- To place the members of the polynomials with descending exponents of the variable.
- To share a senior member of the dividend on the senior member of the divider.
- The result is multiplied by the divisor and subtract this product from the dividend.
- 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: