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.

**Degree- **

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

Lets Work Out-
Solution- The degree of the polynomial is 4. |

**Standard Form- **

The standard form of writing a polynomial is to put the terms with highest degree first then at the last constant term.

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

**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-
= \(7s^{3}+(2s^{2}+5s^{2})+(3s+2s)+(9+1)\) = \(7s^{3}+7s^{2}+5s+10\) Hence addition result in a polynomial
= \(7s^{3}+(2s^{2}-5s^{2})+(3s-2s)+(9-1)\) = \(7s^{3}-3s^{2}+s+8\) Hence addition result in a polynomial
\(= 7s^{3} (5s^{2}+2s+1)+2s^{2} (5s^{2}+2s+1)+3s (5s^{2}+2s+1)+9 (5s^{2}+2s+1)\) \(= (35s^{5}+14s^{4}+7s^{3})+ (10s^{4}+4s^{3}+2s^{2})+ (15s^{3}+6s^{2}+3s)+(45s^{2}+18s+9)\) \(= 35s^{5}+(14s^{4}+10s^{4})+(7s^{3}+4s^{3}+15s^{3})+ (2s^{2}+6s^{2}+45s^{2})+ (3s+18s)+9\) \(= 35s^{5}+24s^{4}+26s^{3}+ 53s^{2}+ 21s +9\)
\(\large \frac{7s^{3}+2s^{2}+3s+9}{5s^{2}+2s+1}\) These cannot be simplified. Therefore division of these polynomial do not result in a Polynomial. |

Learn more about Polynomial and Polynomial Functions.

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