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Similar to Class 10, Maths- Ch-2, Polynomials ppt./ (20)
Class 10, Maths- Ch-2, Polynomials ppt./
- 1. Welcom e
- 2. TABLE OF CONTENTS 01 04 03 02 To Polynomials • On the basis of degree • On the basis of term Zeroes and coefficients of Quadratic polynomials. Number of zeroes a polynomial can have.
- 3. • A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division. • Examples of constants, variables and exponents are as follows: a) Constants Example: 1, 2, 3, etc. b) Variables Example: g, h, x, y, etc. c) Exponents Example: 5 in x5 etc. • Example of a polynomial is 4x2+1x+3.
- 4. TYPES OF POLYNOMIALS ON THE BASIS OF TERMS ON THE BASIS OF DEGREE LINEAR POLYNOMIAL QUADRATIC POLYNOMIAL CONSTANT POLYNOMIAL TRINOMIAL BINOMIAL MONOMIAL CUBIC POLYNOMIAL
- 5. • What is the Degree of a Polynomial ? Ans: Thedegreeofapolynomialisthehighestsumoftheexponents/powersof apolynomial’svariables. Here,inthisexamplethehighest powersofthegivenpolynomialsis thewritteninthedegreecolumn.
- 6. ON THE BASIS OF DEGREE 1. Constant polynomial A polynomial of degree zero is called a ‘Constant Polynomial’. Example – 7x 0 = 7 , 4x 0 = 4, etc. It is also called zero polynomial. 2. Linear polynomial A polynomial of degree 1 is called ‘Linear Polynomial’ Example – 2x-3, 4x+9, etc. The most general form of linear polynomial is ax+b, where a ≠ 0 and a & b are real.
- 7. 3. Quadratic Polynomial A polynomial with degree 2 is called a ‘Quadratic polynomial’. Example – 2x 2 +3x+6, x 2 - 2, etc. Generally, quadratic polynomial with x variable is written in the form of ax2 + bx + c, where a ≠0 and a, b & c are real. 4. Cubic Polynomial A polynomial of degree 3 is called ‘Cubic polynomial’. Example- 2-x3 , x3 – 1,etc. The most general form of cubic polynomials is ax3 +bx2 +cx+1 where a, b, c & d are real.
- 9. Let α , β and γ be the zeroes of the polynomial f(x)= ax²+bx+c. By factor theorem (x- α)and (x- β) are the factors of f(x). ∴ f(x)=k(x- α)(x- β), where k is a constant. Now putting all the values - ⇒ ax²+bx+c= k {x²-(α+ β )x+ αβ} ⇒ax²+bx+c= kx²-k(α+ β )x+ kαβ Comparing the coefficients of x²,x and constant terms on both sides, we get a = k, b = -k(α+ β ) and c = kαβ ⇒ α+ β= -b/a and αβ=c/a ⇒ α+ β(Sum of the zeroes)= -(Coefficient of x) Coefficient of x² and αβ(Product of the zeroes)= Constant term Coefficient of x²
- 10. • The graph of a linear polynomial intersects the x-axis at a maximum of one point. Therefore, a linear polynomial has a maximum of one zero. • The graph of a quadratic polynomial intersects the x-axis at a maximum of two points. Therefore, a quadratic polynomial can have a maximum of two zeroes. • The graph of a cubic polynomial intersects the x- axis at maximum of three points. A cubic polynomial has a maximum of three zeroes.