Slide 1: MATHS PROJECT Polynomials BY- YASH BANSAL 9 TH A
Slide 2: CONTENTS 1.INTRODUCTION 2.GEOMETRICAL MEANING OF ZEROES OF THE POLYNOMIAL 3.RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL 4.DIVISION ALGORITHM FOR POLYNOMIAL 5.SUMMARY 6.QUESTIONS AND EXERCISE
Slide 3: POLYNOMIALS 2x 2 + 3x = 5 2x 2 + 3x= 9 x 3 – 3x 2 + x +1 = 0 4y 3 - 4y 2 + 5y + 8 = 0 9x 2 + 9y + 8 =0
Slide 4: INTRODUCTION In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions.
Slide 5: RECAPTULATION
Slide 6: POLYNOMIAL Let x be a variable n, be a positive integer and as, a 1 ,a 2 ,….a n be constants (real nos.) Then, f(x) = a n x n + a n-1 x n-1 +….+a 1 x+x o a n x n ,a n-1 x n-1 ,….a 1 x and a o are known as the terms of the polynomial. a n ,a n-1 ,a n-2 ,….a 1 and a o are their coefficients . For example : p(x) = 3x – 2 is a polynomial in variable x. q(x) = 3y 2 – 2y + 4 is a polynomial in variable y. f(u) = 1/2u 3 – 3u 2 + 2u – 4 is a polynomial in variable u. NOTE : 2x 2 – 3√x + 5, 1/x 2 – 2x +5 , 2x 3 – 3/x +4 are not polynomials .
Slide 7: The exponent of the highest degree term in a polynomial is known as its degree . For example : f(x) = 3x + ½ is a polynomial in the variable x of degree 1. g(y) = 2y 2 – 3/2y + 7 is a polynomial in the variable y of degree 2. p(x) = 5x 3 – 3x 2 + x – 1/√2 is a polynomial in the variable x of degree 3. q(u) = 9u 5 – 2/3u 4 + u 2 – ½ is a polynomial in the variable u of degree 5.
Slide 8: CONSTANT Polynomial A polynomial of degree zero is called a constant polynomial. LINEAR polynomial A polynomial of degree one is called a linear polynomial For example: f(x) = 7, g(x) = -3/2, h(x) = 2 are constant polynomials. The degree of constant polynomials is not defined. For example: p(x) = 4x – 3, q(x) = 3y are linear polynomials. Any linear polynomial is in the form ax + b, where a, b are real nos. and a ≠ 0. It may be a monomial or a binomial. F(x) = 2x – 3 is binomial whereas g (x) = 7x is monomial.
Slide 9: A polynomial of degree two is called a quadratic polynomial. f(x) = √3x 2 – 4/3x + ½, q(w) = 2/3w 2 + 4 are quadratic polynomials with real coefficients. Any quadratic is always in the form f(x) = ax 2 + bx +c where a,b,c are real nos . and a ≠ 0. QUADRATIC POLYNOMIAL CUBIC POLYNOMIAL A polynomial of degree three is called a cubic polynomial. f(x) = 9/5x 3 – 2x 2 + 7/3x _1/5 is a cubic polynomial in variable x. Any cubic polynomial is always in the form f(x = ax3 + bx2 +cx + d where a,b,c,d are real nos.
Slide 10: VALUE OF POLYNOMIAL ZERO OF THE POLYNOMIAL A real no. x is a zero of the polynomial f(x),is f(x) = 0 Finding a zero of the polynomial means solving polynomial equation f(x) = 0. If f(x) is a polynomial and y is any real no. then real no. obtained by replacing x by y in f(x) is called the value of f(x) at x = y and is denoted by f(x). Value of f(x) at x = 1 f(x) = 2x 2 – 3x – 2 f(1) = 2(1) 2 – 3 x 1 – 2 = 2 – 3 – 2 = -3 Zero of the polynomial f(x) = x 2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0 (x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0 x = -4 , -3
Slide 11: GRAPHS OF THE POLYNOMIALS An nth degree polynomial can have at most n real zeroes. no. of real zeros of a polynomial is less than or equal to degree of the polynomial
Slide 12: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3 CONSTANT FUNCTION DEGREE = 0 MAX. ZEROES = 0
Slide 13: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 LINEAR FUNCTION DEGREE =1 MAX. ZEROES = 1
Slide 14: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 2 + 3x + 2 QUADRATIC FUNCTION DEGREE = 2 MAX. ZEROES = 2
Slide 15: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 3 + 4x 2 + 2 CUBIC FUNCTION DEGREE = 3 MAX. ZEROES = 3
Slide 16: RELATIONSHIP B/W ZEROES AND COEFFICIENTS OF A POLYNOMIAL QUADRATIC A + B = - coefficient of x Coefficient of x 2 = - b a AB = constant term Coefficient of x 2 = c a
Slide 17: RELATIONSHIP B/W ZEROES AND COEFFICIENTS OF POLYNOMIALS CUBIC A + B + C = -Coefficient of x 2 = -b Coefficient of x 3 a AB + BC + CA = Coefficient of x = c Coefficient of x 3 a ABC = - Constant term = d Coefficient of x 3 a
Slide 18: RELATIONSIPS ON VERYFYING THE RELATIONSHIP BETWEEN THE ZEROES AND COEFFICIENTS ON FINDING THE VALUES OF EXPRESSIONS INVOLVING ZEROES OF QUADRATIC POLYNOMIAL ON FINDING AN UNKNOWN WHEN A RELATION BETWEEEN ZEROES AND COEFFICIENTS ARE GIVEN. OF ITS A QUADRATIC POLYNOMIAL WHEN THE SUM AND PRODUCT OF ITS ZEROES ARE GIVEN.
Slide 19: DIVISION ALGORITHM
Slide 20: If f(x) and g(x) are any two polynomials with g(x) ≠ 0,then we can always find polynomials q(x), and r(x) such that : F(x) = q(x) g(x) + r(x), Where r(x) = 0 or degree r(x) < degree g(x) ON VERYFYING THE DIVISION ALGORITHM FOR POLYNOMIALS. ON FINDING THE QUOTIENT AND REMAINDER USING DIVISION ALGORITHM. ON CHECKING WHETHER A GIVEN POLYNOMIAL IS A FACTOR OF THE OTHER POLYNIMIAL BY APPLYING THEDIVISION ALGORITHM ON FINDING THE REMAINING ZEROES OF A POLYNOMIAL WHEN SOME OF ITS ZEROES ARE GIVEN.
Slide 21: made by : YASH BANSAL cLaSs : IX - a SAGAR PUBLIC sCHool