# Polynomials

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.

#### Practise This Question

If I < x < I +1, Find [-x], where I is an integr