Definition:

A polynomial in ‘x’ is an algebraic expression consisting of a single term or a finite sum of terms in the form axn, where ‘a’ is any real number and ‘n’ is a non-negative integer.

Examples:

• 4×5
• x3 - 4. 3×2 + 1
• 5x - 2
• 6×2 + 2x - 7
• 3×3 - 7×2 - 6x + 4

The following are the examples of algebraic expressions which are not polynomials:

• 3 x - 2 - 2x - 3 is not a polynomial because exponents of the variables have to be non-negative integers.
• is not a polynomial as the power (or exponent) of x is a negative integer.

Each term in the polynomial is called a monomial, which consists of a constant term multiplied by one or more variables. Each variable may have a constant positive integer called the exponent. The exponent on the variable is the degree of the variable.

In the term, axn, ‘a’ is called the coefficient of the term and ‘n’ which is the exponent of the term is called the degree of the term. The coefficients take on the sign before the term.

For example, consider the polynomial 4×3 - 3×2 + 7x + 9, the coefficient of x3 is 4, the coefficient of x2 is - 3, the coefficient of x is 7 and 9 is the constant term.

Constant term in the polynomial is a term with no variables preceding or succeeding it.

The constant term is a monomial with degree zero.

Examples: 7, - 22

A polynomial is usually written in the descending order of powers of a variable.

Examples: 8×3 - 3×2 + 1, 2×3 + 3×2 - 5x + 7

Classification of Polynomials according to the number of terms

Monomial: A polynomial with one term is called a Monomial.

Examples: 2x, 7xy, 11×3y6 are some examples of monomials.

Binomial: A polynomial with two terms is called a Binomial.

Examples: 3x + 1, 2×3 - 5x, x4 - 4 are some examples of binomials.

Trinomial: A polynomial with three terms is called a Trinomial.

Examples: 2×2 + 4x - 11, 4×3 - 13x + 9, 7×3 - 22×2 + 24x are some examples of trinomials.

Degree of a Polynomial: The degree of a polynomial in one variable is the highest power of the variable in that polynomial.

The degree of the polynomial 7×3 - 4×2 + 2x + 9 is 3, as the highest power of the variable ‘x’ is 3.

The degree of a monomial with more than one variable is the sum of the exponents (powers) on those variables.

Examples:

• 24xyz is a monomial with degree 3. (Sum of degrees of x, y and z = 1 + 1 + 1 = 3)
• 13×4y2z7 is a monomial with degree 13. (Sum of degrees of x, y and z = 4 + 2 + 7 = 13)

The degree of a polynomial with more than one variable is the degree of the monomial of the highest degree in that polynomial.

Examples:

• Degree of the polynomial 7×3y2 z6 - 4×2y6z - 9x + 15 is 11. (Highest degree is that of the first monomial 3 + 2 + 6 = 11)

Leading Coefficient of a Polynomial: The leading coefficient of the polynomial is the coefficient of the variable with the greatest degree. The leading coefficient of the polynomial 8×2 + 6x - 3 is 8 because it is the coefficient of x2, the variable with highest degree.

Evaluation of a Polynomial: Evaluation of a polynomial consists of assigning a numerical value to each of its variables and carrying out the indicated mathematical operations.

For example:

• Evaluate the polynomial 4×3 + 6 when x = 2

4×3 + 6 = 4 (2)3 + 6

= 4 (8) + 6

= 32 + 6 = 38

• Evaluate the polynomial 5×2 + 3x - 4 when x = 3

5×2 + 3x - 4 = 5 (3)2 + 3 (3) - 4

= 5 (9) + 3 (3) - 4

= 45 + 9 - 4 = 50

Further Reading on Polynomials:

• Addition of Polynomials
• Subtraction of Polynomials
• Multiplication of Polynomials
• Division of Polynomials

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