Polynomial | What is Polynomial?

A polynomial is an algebraic expression of the form a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + …………… + a, where n is a non-negative integer and the coefficients an, a_n-1, a_n-2, ………, a are real numbers with a_n ≠ 0. It is called a polynomial of degree n in variable x.

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Some examples of polynomial are:

-           3x + 5

-           2x² + x – 1

-           X³ + 2x² + 5x + 6

Multinomial and polynomial do not mean the same thing. All polynomials are multinomial, but all multinomial may not be polynomials. For example: 3x³ + 2x² + 6x + 2√x – 9 is a multinomial but not a polynomial.

A polynomial in a variable x is said to be in standard form, if it is in simplified form and arranged in ascending or descending order. A polynomial in variable x is usually denoted by the symbols f(x), g(x), h(x), etc.

The standard form of a polynomial is in the form of:

            f(x)= a_nxⁿ +a_n-1x^n-1 +a_n-2x^n-2 + …………….... +a.

or,        g(x) = a + a₁x + a₂x² + …………….... + a_nxⁿ.

(i)       If a_n ≠ 0, then n is called the degree of the polynomial f(x). Degree of polynomial can never be negative.

(ii)      a_nxⁿ, a_n-1x^n-1, a_n-2x^n-2, ……… , a are called the terms of the polynomial f(x). a is called the constant term.

(iii)     a, a₁, a₂, ……….., a_n are called the coefficients of the polynomial f(x).

(iv)     If a_n ≠ 0, then a_nxⁿ is called the leading term and a_n is called the leading coefficient of the polynomial.

(v)      If all the coefficients a_n, a_n-1, a_n-2, ………. , a are zero, then f(x) is called zero polynomial. The degree of zero polynomial is never defined.

(vi)     The degree of a polynomial is zero if and only if it is a non-zero constant polynomial. For example, f(x) = 5 is a zero degree polynomial.

A polynomial of degree one (1) is called linear polynomial. Polynomials of degree 2, 3 and 4 are respectively called quadratic polynomial, cubic polynomial and biquadratic polynomial. For example:

-           f(x) = ax + b is linear polynomial.

-           f(x) = ax² + bx + c is quadratic polynomial.

-           f(x) = ax³ + bx² + cx + d is cubic polynomial.

-           f(x) = ax⁴ + bx³ + cx² + dx + e is biquadratic polynomial.

A polynomial, non of whose coefficients is zero is called a complete polynomial. Otherwise it is an incomplete polynomial.

If the coefficients of a polynomial are integers, then the polynomial is said to be the polynomial over integers. f(x) = 5x³ – 3x² + 3x + 2 is a polynomial over integers. If the coefficients of a polynomial are rational numbers, it is called the polynomial over rationals. A polynomial is called polynomial over real, if its coefficients are real numbers.

If a polynomial contains two variables x and y, it is regarded as a function of x and y.

For example,

f(xy) = x²y + 5xy + y²

A polynomial containing three variables x, y and z is written as follows:

            F(xyz) = x²yz + x²z + xz² + xy²z + 2.

The degree of polynomial in two or more variables is the highest sum of the powers of the term involved in the polynomial.

For example:

-           f(xy) = 3xy³ – x² + 5x – 8 is a polynomial of degree 4.

-           f(xy) = 5x³y⁴ is a polynomial of degree 7.

Equal polynomials

Two polynomials are said to be equal if and only if

(i)       their degree are same

(ii)      coefficients of corresponding terms are same.

If f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + …………… + a₀ and g(x) = b_mx^m + b_m-1x^m-1 + b_m-2x^m-2 + …………… + b₀ are equal polynomials, then n = m, a_n = b_m, a_n-1 = b_m-n, …………… , a₀ = b₀.

If polynomials f(x) and g(x) are equal, then we write f(x) = g(x).

Addition and subtraction of polynomials

Let f(x) and g(x) be two polynomials given as follows:

            f(x)= a₀ +a₁x + a₂x² + …………... + a_nxⁿ

            g(x)= b₀ + b₁x + b₂x² + …………... + b_nxⁿ

Then their sum is defined as

f(x) + g(x) = (a₀ + b₀) + (a₁ + b₁)x + (a₂ + b₂)x² + ………… + (a_n + b_n)xⁿ.

Thus the sum of two polynomials can be found by grouping the power terms, retaining their signs and adding the coefficients of like powers.

For the calculation of sum or difference of polynomials, the polynomials are first kept in standard form. Then, the coefficients of like terms are added for sum and subtracted for difference.

Example: Let, f(x) = 3x² + 5x + 6 and g(x) = 4x² – 2x + 1, then

           -    f(x) + g(x) = (3 + 4)x² + (5 – 2)x + (6 + 1)

                    = 7x² + 3x + 7

           -    f(x) – g(x) = (3 – 4)x² + (5 + 2)x + (6 – 1)

                   = -x² + 7x + 5

Properties of addition of polynomials

1.  Closure property: The sum of two polynomials over real numbers is also a polynomial over real numbers. This is called the closure property of addition. i.e. polynomials are closed under addition.

2.  Commutative property: For any two polynomials f(x) and g(x),

f(x) + g(x) = g(x) + f(x).

          This is called the commutative property of addition.

3.  Associative property: For any three polynomials f(x), g(x) and h(x)

f(x) + [g(x)+h(x)] = [f(x)+g(x)] + h(x)

          This is called the associative property of addition.

4.  Additive identity: For any polynomial f(x), there is a polynomial 0 such that

f(x) + 0 = f(x)

          Then 0 is called the additive identity.

5.  Additive inverse: For any of the polynomial f(x) there is a polynomial –f(x) such that

f(x) + [-f(x)] = 0, the additive identity.

            Then –f(x) is called additive inverse of f(x).

You can comment your questions or problems regarding the polynomials here.