Slide 1: Maths is my favourite subject I love maths MATHS PROJECT Polynomials
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: ACKNOWLEDGEMENT 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 for their help, encouragement and blessings. 1234567891011121314151617181920
Slide 4: POLYNOMIALS 2x2 + 3x = 5 2x2 + 3x= 9 x3 – 3x2 + x +1 = 0 4y3 - 4y2 + 5y + 8 = 0 9x2 + 9y + 8 =0
Slide 5: 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 6: RECAPTULATION
Slide 7: POLYNOMIAL 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.
Slide 8: DEGREE OF A POLYNOMIAL 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.
Slide 9: 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 10: 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. QUADRATIC POLYNOMIAL CUBIC POLYNOMIAL 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.
Slide 11: 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) = 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
Slide 12: 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 13: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3
CONSTANT FUNCTION
DEGREE = 0
MAX. ZEROES = 0
Slide 14: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2
LINEAR FUNCTION
DEGREE =1
MAX. ZEROES = 1
Slide 15: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2
QUADRATIC FUNCTION
DEGREE = 2
MAX. ZEROES = 2
Slide 16: GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2
CUBIC FUNCTION
DEGREE = 3
MAX. ZEROES = 3
Slide 17: RELATIONSHIP B/W ZEROES AND COEFFICIENTS OF A POLYNOMIAL QUADRATIC A + B = - coefficient of x Coefficient of x2 = - b a AB = constant term Coefficient of x2 = c a
Slide 18: RELATIONSHIP B/W ZEROES AND COEFFICIENTS OF POLYNOMIALS CUBIC 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
Slide 19: 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 20: DIVISION ALGORITHM
Slide 21: 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 22: made by : Arshika Singh cLaSs : X - a IndIrAPUram puBliC sCHool Teacher Incharge: niVeDIta SaxeNa