Polynomials

Polynomials are expressions which are composed of two algebraic terms. Here, a 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.

Polynomial

Polynomials Introduction

Polynomial is made up of two terms namely Poly (meaning “many”) and Nomial (meaning “terms.”).

A polynomial 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 –

\(\frac{1}{x+2}, x^{-3}\)

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.

What is the Degree of a Polynomial?

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

Lets Work Out-

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

Solution- The degree of the polynomial is 4.

Polynomials Related Concepts

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 give below:

\(a_{0}x^{n}+a_{1}x^{n-1}+ a_{2}x^{n-2}+ ……..+ a_{n-2}x^{2}+a_{n-1}x + a_{n}\)

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.

 

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

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

These 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

Degree of a constant polynomial is