CHAPTER.ppt of polinomial chapter 2 class 9 ppt ppt ppt

polinomial

1. CHAPTER POLYNOMIALS INTRODU CTION :- An expression of more than two algebraic terms, especially the of several terms that contain different powers of the same variables .

2. INTRODUCTION OF POLYNOMIALS For the purpose of this definition, consider a polynomial as an expression that is made up of variables and constants, and whose exponents of the variables are only positive integer numbers rather than fractions. The polynomial terms are mostly separated by addition or subtraction operators, with a few exceptions. Polynomials are used in a variety of applications, including the formulation of polynomial equations and the definition of polynomial functions .  For example, the polynomial 5×2 – x + 1 can be written as It is not a polynomial in the algebraic expression 3×3 + 4x + 5/x + 6×3/2 because one of the powers of ‘x,’ the power of 3, and the power of 4, are fractions, and the other is negative. Polynomials are expressions that contain one or more terms that have a non-zero coefficient in their coefficients. Variables, exponents, and constants are included among the terms. The leading term of a polynomial is the first term that appears in the polynomial.

3. Types of polynomials based on degree : The degree of a polynomial is defined as the highest power of the leading term or the highest power of the variable in the polynomial. This is achieved by arranging the polynomial terms in the descending order of their powers before multiplying them together. They can be divided into four major types based on the degree of the polynomial being considered . They are, in fact. Zero or Constant polynomial Linear polynomial Quadratic polynomial Cubic polynomial

4. Types of polynomials Meaning Examples Zero or constant polynomial A zero polynomial is a polynomial that has exactly zero degrees. 3 or 3×0 Linear polynomial Linear polynomials are polynomials that have the degree of one as the degree of the polynomial. The highest exponent of the variable(s) in a linear polynomial is 1, and the lowest exponent is 0. x + y – 4, 5m + 7n, 2p Quadratic polynomial Quadratic polynomials are polynomials with the degree of 2 as the degree of the polynomial in question. 8×2 + 7y – 9, m2 + mn – 6 Cubic polynomial Cubic polynomials are polynomials that have the number three as the degree of the polynomial. 3×3,   p3 + pq + 7

5. Types of polynomials based on terms: There are several different types of polynomials based on the number of terms they contain. There are polynomials with one term, polynomials with two terms, polynomials with three terms, and even more terms. Polynomials are classified into the following categories based on the number of terms Monomials : When a polynomial expression contains only one term, it is referred to as a monomial expression. For instance, 4t, 21x, 2y, and 9pq. Apart from that, the sum of 2x, 5x, and 10x is a monomial because these are like terms that when added together result in 17x. Binomials : In contrast to terms, a binomial is a polynomial with only two terms. For example, 3x + 4×2 is a binomial because it contains two unlike terms, namely 3x and 4×2, and 10pq + 13p2 is a binomial because it contains two unlike terms, namely 10pq and 13p2. Trinomials : In contrast to terms, a trinomial is a polynomial with three factors. For example, 3x + 5×2 – 6×3 and 12pq + 4×2 – 10 are both trinomials.  polynomial expression can contain more than three terms at the same time. Four-term polynomials are polynomials that have four terms that are unlike each other. Polynomials with 5 terms , for example, are referred to as five-term polynomials, and so forth.

6. Zeros of Polynomials Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The  degree of a polynomial  is the highest power of the variable x Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The  degree of a polynomial  is the highest power of the variable x. A polynomial of degree 1 is known as a linear polynomial. The standard form is ax + b, where a and b are real numbers and a≠0. 2x + 3 is a linear polynomial. A polynomial of degree 2 is known as a quadratic polynomial. Standard form is ax 2  + b x + c, where a, b and c are real numbers and a ≠ 0 x 2 + 3x + 4 is an example for quadratic polynomial. Polynomial of degree 3 is known as a cubic polynomial. Standard form is ax 3 + bx 2  + cx + d, where a, b, c and d are real numbers and a≠0. x 3  + 4x + 2 is an example for cubic polynomial. Zero of Polynomial K = -(Constant/ Coefficient of x)

7. Remainder Theorem Definition The Remainder Theorem begins with a polynomial say p(x), where “p(x)” is some polynomial p whose variable is x. Then as per theorem, dividing that polynomial p(x) by some linear factor x – a, where a is just some number. Here go through a long polynomial division,  which results in some polynomial q(x) (the variable “q” stands for “the quotient polynomial”) and a polynomial remainder is r(x). It can be expressed as: p(x)/x-a = q(x) + r(x) That is when we divide p(x) by x-a we obtain p(x ) = (x-a)·q(x) + r(x), as we know that  Dividend = (Divisor × Quotient) + Remainder But if r(x) is simply the constant r (remember when we divide by (x-a) remainder is a constant)…. so we obtain the following solution, i.e p(x ) = (x-a)·q(x) + r Observe what happens when we have x equal to a: p(a) = (a-a)·q(a) + r p(a ) = (0)·q(a) + r p(a ) = r Hence , proved.

8. Factor Theorem Factor Theorem  is generally applied to factoring and finding the roots of polynomial equations. It is the reverse form of the remainder theorem. Problems are solved based on the application of synthetic division and then to check for a zero remainder. When  p(x) = 0  then  y-x  is a factor of the polynomial Or if we consider the other way, then When  y-x  is a factor of the polynomial then  p(x)  =0

9. An Algebraic Expression An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms. Also, solve questions in  Algebraic Expressions For eg . (5x – 3), x is a  variable , whose value is unknown to us which can take any value. 5 is known as the c oefficient  of x, as it is a constant value used with the variable term and is well defined. 3 is the c onstant  value term that has a definite value. The whole expression is known to be the Binomial term, as it has two unlikely terms.

10. Algebraic Identitys Identity I:  (a + b) 2  = a 2  + 2ab + b 2 Identity II:   (a – b) 2  = a 2  – 2ab + b 2 Identity III:  a 2  – b 2 = (a + b)(a – b) Identity IV:  (x + a)(x + b) = x 2  + (a + b) x + ab Identity V:  (a + b + c) 2  = a 2  + b 2  + c 2  + 2ab + 2bc + 2ca Identity VI:  (a + b) 3  = a 3  + b 3  + 3ab (a + b) Identity VII:   (a – b) 3  = a 3  – b 3  – 3ab (a – b) Identity VIII:  a 3  + b 3  + c 3  – 3abc = (a + b + c)(a 2  + b 2  + c 2  – ab – bc – ca)