Class 9 Maths Polynomials 2.pptx ppt ppt

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1. B.J.P.S Samiti’s M.V.HERWADKAR ENGLISH MEDIUM HIGH SCHOOL CLASS: 9 th CHAPTER 2: POLYNOMIALS Program: Semester: Course: NAME OF THE COURSE Staff Name: VINAYAK PATIL 1

2. POLYNOMIALS (1) Polynomial : The expression which contains one or more terms with non-zero coefficient is called a polynomial. A polynomial can have any number of terms. For Example : 10, a + b, 7x + y + 5, w + x + y + z, etc. are some polynomials (2) Degree of polynomial : The highest power of the variable in a polynomial is called as the degree of the polynomial. For Example : The degree of p(x) = x 5  – x 3  + 7 is 5.

3. Types of Polynomials based on terms 1. Monomials  – Monomials are the algebraic expressions with one term, hence the name “ Mono”mial . In other words, it is an expression that contains any count of like terms. For example, 2x, 4t, 21x 2 y, 9pq etc . 2. Binomials  – Binomials are the algebraic expressions with two unlike terms, hence the name “ Bi”nomial . For example, 3x + 4x 2 , 10pq + 13p 2 q 3. Trinomials  – Trinomials are the algebraic expressions with three unlike terms, hence the name “ Tri”nomial . For example- 3x + 5x 2  – 6x 3

4. Types of Polynomials based on degree(powers) (1) Linear polynomial : A polynomial of degree one is called a linear polynomial. In general a linear polynomial can be expressed in the form ax + b where a≠0 For Example : 2x – 7, x + 5, etc. are some linear polynomial. (2) Quadratic polynomial : A polynomial having highest degree of two is called a quadratic polynomial. The term ‘quadratic’ is derived from word ‘quadrate’ which means square. In general, a quadratic polynomial can be expressed in the form ax 2  + b x + c, where a≠0 and a, b, c are constants. For Example : x 2  – 9, a 2  + a + 7, etc. are some quadratic polynomials. (3) Cubic Polynomial : A polynomial having highest degree of three is called a cubic polynomial. In general, a cubic polynomial can be expressed in the form ax 3  + bx 2  + cx + d, where a≠0 and a, b, c, d are constants. For Example : x 3  – 9x +2, a 3  + a 2  + a + 7, etc. are some cubic polynomial.

5. Zeroes of a Polynomial Zeroes of a Polynomial : The value of variable for which the polynomial becomes zero is called as the zeroes of the polynomial. In general, if k is a zero of p(x) = ax + b, then p(k) = a k + b = 0, i.e., k = - b/a. Hence, the zero of the linear polynomial ax + b is –b/a = -(Constant term)/(coefficient of x) For Example : Consider p(x) = x + 2. Find zeroes of this polynomial. If we put x = -2 in p(x), we get, p(-2) = -2 + 2 = 0. Thus, -2 is a zero of the polynomial p(x).

6. A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0. Every linear polynomial in one variable has a unique zero, a non – zero constant polynomial has no zero, and every real number is a zero of the polynomial.

7. Factorisation of Polynomials Factor theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number then i ) x – a is a factor of p(x), if p(a) = 0 and ii) p(a) = 0, if x – a is a factor of p(x)

8. Algebraic Identities