This is ppt of chapter 2 NCERT Polynomials

Published on: **Mar 4, 2016**

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Science

- 1. The Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
- 2. Introduction of Polynomials Polynomials = Poly (means many) + nomials (means terms). Thus, a polynomial contains many terms Thus, a type of algebraic expression with many terms having variables and coefficients is called polynomial. Example – Let us consider third example, in this ‘x’ is called variable. Power of ‘x’, i.e. 2 is called exponent. Multiple of ‘x’, i.e. 2 is called coefficient. The term ‘2’ is called constant. And all items are called terms.
- 3. Let us consider the second example – In this there are two variables, i.e. x and y. Such polynomials with two variables are called Polynomials of two variables Power of x is 2. This means exponent of x is 2. Power of y is 1. This means exponent of y is 1. The term ‘5’ is constant. There are three terms in this polynomial.
- 4. Types of Polynomial: Monomial – Algebraic expression with only one term is called monomial. Example – Binomial – Algebraic expression with two terms is called binomial. Example – Trinomial – Algebraic expression with three terms is called trinomial. Example – But algebraic expressions having more than two terms are collectively known as polynomials.
- 5. Variables and polynomial: Polynomial of zero variable If a polynomial has no variable, it is called polynomial of zero variable. For example – 5. This polynomial has only one term, which is constant. Polynomial of one variable – Polynomial with only one variable is called Polynomial of one variable. Example – In the given example polynomials have only one variable i.e. x, and hence it is a polynomial of one variable.
- 6. Polynomial of two variables – Polynomial with two variables is known as Polynomial of two variables. Example – In the given examples polynomials have two variables, i.e. x and y, and hence are called polynomial of two variables. Polynomial of three variables – Polynomial with three variables is known as Polynomial of three variables. Example –
- 7. Degree of Polynomials: Highest exponent of a polynomial decides its degree. Polynomial of 1 degree: Example: 2x + 1 In this since, variable x has power 1, i.e. x has coefficient equal to 1 and hence is called polynomial of one degree. Polynomial of 2 degree – Example: In this expression, exponent of x in the first term is 2, and exponent of x in second term is 1, and thus, this is a polynomial of two(2) degree. To decide the degree of a polynomial having same variable, the highest exponent of variable is taken into consideration. Similarly, if variable of a polynomial has exponent equal to 3 or 4, that is called polynomial of 3 degree or polynomial of 4 degree respectively.
- 8. Important points about Polynomials: A polynomial can have many terms but not infinite terms. Exponent of a variable of a polynomial cannot be negative. This means, a variable with power - 2, -3, -4, etc. is not allowed. If power of a variable in an algebraic expression is negative, then that cannot be considered a polynomial. The exponent of a variable of a polynomial must be a whole number. Exponent of a variable of a polynomial cannot be fraction. This means, a variable with power 1/2, 3/2, etc. is not allowed. If power of a variable in an algebraic expression is in fraction, then that cannot be considered a polynomial. Polynomial with only constant term is called constant polynomial. The degree of a non-zero constant polynomial is zero. Degree of a zero polynomial is not defined.