Mathematics Polynomials and is different types

10th Standard Mathematics Polynomials

1. Algebraic Expressions An algebraic expression is an expression made up of variables and constants along with mathematical operators. An expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them. E.g . x + 5y – 10 , 2x + 1, x + y

2. Polynomials What is the meaning of word polynomial ? The word polynomial is derived from the Greek words ‘poly’ means ‘many‘ and ‘nominal’ means ‘terms‘, so altogether it is said as “many terms”

3. What is a Polynomial? An algebraic expression, if the exponent /power of the variables are whole numbers then that algebraic expression is called as polynomial. 6, -8, , 0, etc. are constant numbers and called as constant polynomials. Remember that , power of variable should not be a fraction or negative number.  

4. Here, in this example, power of all variables is either 1 or 2 i.e. whole number so given algebraic expression in a polynomial.

5. Which of the following algebraic expressions are polynomials? Sr. No. Algebraic expression Is it polynomial ? (Yes/No) 1 2 3 4 5 y +   No 7 x +9   2 - 5   2 7 m – 6   11 No No Yes Yes

6. Classification of polynomials We classify the polynomials mainly by two ways Number of terms and Degree of polynomial

7. Number of terms 6x 4 + 3x 2 + 5x +19 here in this polynomial there are 4 terms.

8. How many terms are there in the polynomials? Sr. No. Polynomial Number of terms 1 2 3 4 5 1 + +5   3 + 7 x +9     2 7 m   2 2 4

9. Degree of a Polynomial (when only one variable is present) If given polynomial contain only one variable then its degree is the highest power/ exponent of the variable in the given polynomial. e.g. 1) The degree of the polynomial x 2 + 2x+3 is 2, as the highest power of x in the given expression is 2. 2) the degree of the polynomial x 8  + 2x 6  – 3x + 9 is 8 , since the greatest power in the given expression is 8 .

10. Note that, Degree of any non -zero constant polynomial i.e. Any real number except 0 is zero. Why??? I can write any number let's say 5 as 5 , here power of variable is 0 , ∴ Degree is 0. And such polynomials are called as constant polynomials. 0 is constat number and called as Zero polynomial . Que. What is the degree of zero polynomial? Not defined .  

11. Degree of a Polynomial (when more tha n one variable are present) If given polynomial contains more than one variables, them its degree is the highest sum of power of variables in each term of the polynomial is the degree of polynomial. e.g. 3 + 7 − mn , is a polynomial in two variables m and n. Here sum of power of variables in first terms (3 ) is 4+5 = 9 sum of power of variables in second terms (7 ) is 2+3 = 5 sum of power of variables in third terms ( mn ) is 1+1 = 2 ∴ Degree of the given polynomial is 9  

12. Write the degree of following polynomials ? Sr. No. Polynomial Degree 1 2 3 4 5 + +5   5 y + 7 x +9     2 7 m   19 1 3 6

13. Thank you