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-
Example- Find the degree of the polynomial \(6s^{4}+ 3x^{2}+ 5x +19 \) 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-
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. |
Learn more about Polynomial and Polynomial Functions.
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