University Mathematics Handbook (2015)

X. Algebra

Chapter 3. Polynomials

3.1  Definition

a.  , is an -th-degree polynomial. It is denoted .

b.  If , it is a real polynomial.

c.  If , it is a complex polynomial.

d.   are coefficients of the polynomial, is the leading coefficient.

e.  A constant different than zero is a zero-degree polynomial.

f.  A polynomial with all its coefficients equal to zero is called a zero polynomial. A zero polynomial has no degree.

3.2  Sum and Product of Polynomials

Let and be two polynomials.

a.  Polynomials and are equal if and only if and for all .

b.  We define the sum of two polynomials as a collection of like terms.

For example: If , then when

c.  The degree of a sum of polynomials is smaller than or equal to the maximum of their degrees: .

d.  The product of two polynomials is , when (the sum is by every and such that ).

e.  The degree product of polynomials equals to the sum of their degrees:

.

3.3  Polynomial Division

a.  The division of polynomial by polynomial , is represented the following way:

or (*)

when is the quotient, is the remainder, and .

If , then polynomial is divided without a remainder by .

b.  For every two polynomials and there exists a unique pair of polynomials and holding (*) and or .

c.  For the division of polynomials we use the same algorithm as for the division of real numbers.

3.4  Remainder Theorem

a.  Remainder after the division of polynomial by equals to the value of polynomial at , that is .

b.  Polynomial is divided by without remainder, if and only if .

c.  Number zeroing the polynomial, , is called root of the polynomial.

3.5  Factorization of Polynomials

a.  Every polynomial, real or complex, of a degree greater than zero, has at least one complex root.

b.  Every complex polynomial, has exactly roots, not necessarily different, , and

c.  If polynomial is divisible without remainder by , , and indivisible by , then is called the multiplicity of root and is called root with multiplicity .

d.   is the root of polynomials with multiplicity , , if, and only if,

.

e.  If is the root of polynomial with real coefficients, then is also a root of it.

3.6  Vieta's Formulas

Vieta's formulas connect the roots and coefficients of polynomial.

If are roots of polynomial , then: