z : z MDVM (Parle) School Session : 2011-2012 Submitted By : -
Aditya Rathore
Slide 2: Contents INTRODUCTION
GEOMETRICAL MEANING OF ZEROS OF POLYNOMIAL
RELATION BETWEEN ZEROS AND COEFFICIENTS OF A POLYNOMIAL
DIVISION ALGORITHM FOR POLYNOMIALS
SUMMARY
QUESTION AND EXERCISE
Slide 3: Polynomials 3x2+2x=9 X3-3x2+9x-1=5 2x3-3x2+7x-5=0 3x5+5x4+x-2=3 9x2+3y=4
Slide 4: 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. Introduction :
Slide 5: Cont… 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+x0 anxn , an-1xn-1 …….a1x and a0 are known as the terms of polynomial.
an, an-1, an-2……a1 and a0 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 6: Degree of 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) = 9u 5 – 2/3u 4 + u 2 – ½ is a polynomial in the variable u of degree 5.
Slide 7: Constant
Polynomial: A polynomial of degree zero is called a constant polynomial. For example:
f(x) = 7, g(x) = -3/2, h(x) = 2
are constant polynomials.
The degree of constant polynomials is not defined
Slide 8: Linear polynomial: A polynomial of degree one is called a linear polynomial.
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: Types of
polynomial: QUADRATIC POLYNOMIALS 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. CUBIC POLYNOMIALS 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’s & zero’s of Polynomial : 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 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.
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 POLYNOMIALS
Slide 12: GENERAL SHAPES OF POLYNOMIAL : 1) f(x) = 3
CONSTANT FUNCTION
DEGREE = 0
MAX. ZEROES = 0 1 Y -Y -X X 0
Slide 13: Cont…. 2) f(x) = x + 2
LINEAR FUNCTION
DEGREE = 1
MAX. ZEROES = 1 Y -X X -Y 0
Slide 14: Cont…. 3) f(x) = x 2 + 3x + 2
QUADRATIC FUNCTION
DEGREE = 2
MAX. ZEROES = 2
Slide 15: Cont…. 4) f(x) = x 3 + 4x 2 + 2
CUBIC FUNCTION
DEGREE = 3
MAX. ZEROES = 3
Slide 16: RELATIONSHIP B/W ZEROES AND COEFFICIENT OF A POLYNOMIAL
Slide 17: QUADRATIC : α + β = - Coefficient of x
Coefficient of x2
= - b
a _______________ _ αβ = Constant term
Coefficient of x2
= c
a _________________ _
Slide 18: RELATIONSHIPS : ON VERYFYING THE RELATIONSHIP BETWEEN THE ZEROES AND COEFFICIENTS. ONFINDING 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. Cont…. 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: Thanks for being Patient…