Maths

Published on: **Mar 4, 2016**

Published in:
Engineering

- 1. Polynomials
- 2. 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
- 3. The national curriculum framework such that children's life at school must be linked to their life outside the school. this principle marks a de portable use from the legacy of bookish learning and thus the students have been given provisions to preface some project reports on certain subjects. I express my hearty gratitude to CBSE for providing such an interesting and board scope topic for our project. I am really thankful to our respected subject teacher Ms.Nivedita Saxena who helped us in a passive way. I would also like to thank my parents and my friends 123456789101112 131415161718192 0
- 4. 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.
- 5. Let x be a variable n, be a positive integer and as, a1,a2,….an be constants (real nos.) Then, f(x) = anxn+ an-1xn-1+….+a1x+xo anxn,an-1xn-1,….a1x and ao are known as the terms of the polynomial. an,an-1,an-2,….a1 and ao are their coefficients. For example: • p(x) = 3x – 2 is a polynomial in variable x. • q(x) = 3y2 – 2y + 4 is a polynomial in variable y. • f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u. NOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.
- 6. 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) = 2y2 – 3/2y + 7 is a polynomial in the variable y of degree 2. p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the variable x of degree 3. q(u) = 9u5 – 2/3u4 + u2 – ½ is a polynomial in the variable u of degree 5.
- 7. 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.
- 8. A polynomial of degree two is called a quadratic polynomial. f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4 are quadratic polynomials with real coefficients. Any quadratic is always in the form f(x) = ax2 + bx +c where a,b,c are real nos. and a ≠ 0. A polynomial of degree three is called a cubic polynomial. f(x) = 9/5x3 – 2x2 + 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.
- 9. 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) = 2x2 – 3x – 2 f(1) = 2(1)2 – 3 x 1 – 2 = 2 – 3 – 2 = -3 Zero of the polynomial f(x) = x2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0 (x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0 x = -4 , -3
- 10. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3 CONSTANT FUNCTION DEGREE = 0 MAX. ZEROES = 0
- 11. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 LINEAR FUNCTION DEGREE =1 MAX. ZEROES = 1
- 12. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 QUADRATIC FUNCTION DEGREE = 2 MAX. ZEROES = 2
- 13. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 CUBIC FUNCTION DEGREE = 3 MAX. ZEROES = 3
- 14. ☻ A+B =- coefficient of x Coefficient of x2 = - b a ☻ AB = constant term Coefficient of x2 = c a
- 15. A+ B + C = -Coefficient of x2 = -b Coefficient of x3 a AB + BC + CA = Coefficient of x = c Coefficient of x3 a ABC = - Constant term = d Coefficient of x3 a
- 16. 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.