Polynomials

In this section, detailed overview of polynomials are given like its definition, types of polynomials, the standard form of a polynomial, and operations with polynomials along with various solved examples. It should be noted that polynomials is an integral part of the algebra concept and a lot of higher level maths concepts include polynomial fundamentals.

Polynomials

Polynomials Definition

Polynomials are expressions which are composed of two algebraic terms. Polynomial is made up of two terms namely Poly (meaning “many”) and Nominal (meaning “terms.”).

Polynomial equations can have many terms which consist of:

  • Constants such as 1, 2, 3 etc.
  • Variables such as g, h, x, y etc.
  • Exponents such as 5 in \(x^{5}\) etc.

which can be combined using addition, subtraction, multiplication and division but is never division by a variable.

Non Polynomial examples : \(\frac{1}{x+2}, x^{-3}\)

Polynomial Function

A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If a’s are real numbers, then function with one variable and of degree n can be written as:

f(x) = \(a_0x^n + a_1x^{n−1}+a_2x^{n−2}+…..+a_{n−2}x^2+a_{n−1}x + a_n\)

Standard Form of Polynomials

The standard form of writing a polynomial function is to put the terthe ms with highest degree first then, at the last, the constant term. The standard form of any polynomial is given below:

\(a_0x^n+a_1x^{n−1}+a_2x^{n−2}+…….+a_{n−2}x^2+a_{n−1}x+a_n\)

What is the Degree of a Polynomial?

A polynomial having one variable which has the largest exponent is called a degree of the polynomial.

Polynomials Related Concepts

Lets Work Out-

Example- Find the degree of the polynomial \(6s^{4}+ 3x^{2}+ 5x +19 \)

Solution- The degree of the polynomial is 4.

Special types of Polynomial-

  • Monomial- having only one term. Eg. 5x, 3, \(6a^{4}\) etc.
  • Binomial- having two terms. Eg. 5x+3, \(6a^{4} + 17x\)
  • Trinomial- having three terms. Eg. \(8a^{4}+2x+7\) etc.

Operations with Polynomial

  • Addition/Subtraction of polynomials- the addition/subtraction of two or more polynomial always result in a polynomial
  • Multiplication of Polynomial- Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial).
  • Division of Polynomial- Division of two polynomial may or may not result in a polynomial.

Polynomial Examples:

Lets Work Out-

Example- Given two polynomial \(7s^{3}+2s^{2}+3s+9\) and \(5s^{2}+2s+1\). Solve these using mathematical operation.

Solution-

Given polynomial- \(7s^{3}+2s^{2}+3s+9\) and \(5s^{2}+2s+1\)

Addition of Polynomials- \((7s^{3}+2s^{2}+3s+9) + (5s^{2}+2s+1) \)

= \(7s^{3}+(2s^{2}+5s^{2})+(3s+2s)+(9+1)\)

= \(7s^{3}+7s^{2}+5s+10\)

Hence addition result in a polynomial

Subtraction of Polynomials- \((7s^{3}+2s^{2}+3s+9) – (5s^{2}+2s+1) \)

= \(7s^{3}+(2s^{2}-5s^{2})+(3s-2s)+(9-1)\)

= \(7s^{3}-3s^{2}+s+8\)

Hence addition result in a polynomial

Multiplication of Polynomial- \((7s^{3}+2s^{2}+3s+9) \times (5s^{2}+2s+1)\)

\(= 7s^{3} (5s^{2}+2s+1)+2s^{2} (5s^{2}+2s+1)+3s (5s^{2}+2s+1)+9 (5s^{2}+2s+1)\)

\(= (35s^{5}+14s^{4}+7s^{3})+ (10s^{4}+4s^{3}+2s^{2})+ (15s^{3}+6s^{2}+3s)+(45s^{2}+18s+9)\)

\(= 35s^{5}+(14s^{4}+10s^{4})+(7s^{3}+4s^{3}+15s^{3})+ (2s^{2}+6s^{2}+45s^{2})+ (3s+18s)+9\)

\(= 35s^{5}+24s^{4}+26s^{3}+ 53s^{2}+ 21s +9\)

Division of Polynomial- \((7s^{3}+2s^{2}+3s+9) \div (5s^{2}+2s+1)\)

\(\large \frac{7s^{3}+2s^{2}+3s+9}{5s^{2}+2s+1}\)

This cannot be simplified. Therefore division of these polynomial do not result in a Polynomial.

Additional Polynomials Related Articles

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Practise This Question

If a(x)2+bx+c is a monomial in x , what can be said about a, b and c?