POLYNOMIAL - DEFINITION, DEGREE, GENERAL FORM, EXAMPLES, LINKS TO FURTHER STUDY

Polynomial in Basic Algebra,
if you have not already done so.

There, we cover the basics under

The knowledge of various terms
discussed there such as Variables
and Constants, Coefficient, Like
and Unlike terms, Multinomial,
Value of an Algebraic Expression,
Equation, Solution or Root of an Equation
etc is a prerequisite here and in
Algebra Factoring.

Definition :

An algebraic expression in which the exponents of the variable(s) involved are whole numbers (zero or positive integers), is called a Polynomial.

General form :

An algebraic expression of the form
a + bx + cx2 + dx3 + ............ is calleda polynomial in single variable x, briefly a Polynomial.Here, a, b, c, d, .........are constants which are real numbers.
If all the coefficients a, b, c, d, ......... are zero, the expression becomes zero and is called zero polynomial.
If all the coefficients b, c, d, ......... other than the constant term a are zero, the expression becomes a constant (= a)and is called constant polynomial.

Please note that, for an Algebraic Expression to be called Polynomial, the exponents of the variable(s) have to be whole numbers (zero or positive integers) only. If the exponents of the variable(s) are negative integers or fractions or any real numbers other than whole numbers, then it is called multinomial.

All Polynomials are multinomials, but all multinomials are not Polynomials.

Look at the examples for multinomials which are not Polynomials, given under the heading Algebraic Expressions in Basic Algebra, if you have not already done so.

Degree :

The highest exponent of the variable in an Expression is called its Degree.

Example 1 :

Find the degree of the Expressionss
(i) 1 + 2x + 3x2 (ii) y + y3 (iii) 3 + 6z2 + 9z4

Solution:
(i) Degree of 1 + 2x + 3x2 is 2.
(ii) Degree of y + y3 is 3.
(iii) Degree of 3 + 6z2 + 9z4 is 4.

Example 2 :

Write the general form of Expression of
(i) first degree (ii) second degree (iii) nth degree.

Solution:
(i) The required general form is ax + b
where a and b are real number coefficients.
(ii) The required general form is ax2 + bx + cwhere a, b and c are real number coefficients.
This is also called quadratic expression.
(iii) The required general form is
anxn + an-1xn-1 + an-2xn-2 + .....................+a2x2 + a1x + a0
where an, an-1, an-2,.......a2, a1, a0 are real number coefficients.

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Degree when there are two or more variables:

The degree of any term of an Expression with two or more varibles
is the sum of the exponents of all the variables in that term.

Example 3 :

Write the degree of each term in the expression
2x + 3y2 + 4xy + 3x2y3,

Solution:
degree of the term 2x = exponent of x = 1
degree of the term 3y2 = exponent of y = 2
degree of the term 4xy = exponent of x + exponent of y = 1 + 1 = 2
degree of the term 3x2y3 = exponent of x + exponent of y = 2 + 3 = 5.

The Degree of Expression with two or more varibles is the degree of the highest degree term in it.

Example 4 :

Write the degree of the expression in Example 3.

Solution:
In the above example (i.e. in the expression 2x + 3y2 + 4xy + 3x2y3), degree of the expression = degree of the highest degree term in it = 5.

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