Polynomials

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Polynomials

  1. 1. 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.
  2. 2. 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.
  3. 3. •A polynomial of degree 1 is called a Linear Polynomial. Its general form is ax+b where a is not equal to 0 •A polynomial of degree 2 is called a Quadratic Polynomial. Its general form is ax3+bx2+cx, where a is not equal to zero •A polynomial of degree 3 is called a Cubic Polynomial. Its general form is ax3+bx2+cx+d, where a is not equal to zero. •A polynomial of degree zero is called a Constant Polynomial
  4. 4. 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. QUADRATIC POLYNOMIAL For example: f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4 are quadratic polynomials with real coefficients.
  5. 5. For example: If(x) = 7, g(x) = -3/2, h(x) = 2 are constant polynomials. CUBIC POLYNOMIAL CONSTANT POLYNOMIAL For example: f(x) = 9/5x3 – 2x2 + 7/3x _1/5 is a cubic polynomial in variable x.
  6. 6. 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 POLYNOMIAL ZEROES OF 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.
  7. 7. 1. f(x) = 3 CONSTANT FUNCTION DEGREE = 0 MAX. ZEROES = 0
  8. 8. 2. f(x) = x + 2 LINEAR FUNCTION DEGREE =1 MAX. ZEROES = 1
  9. 9. 3. f(x) = x2 + 3x + 2 QUADRATIC FUNCTION DEGREE = 2 MAX. ZEROES = 2 These curves are also called as parabolas
  10. 10. 4. f(x) = x3 + 4x2 + 2 CUBIC FUNCTION DEGREE = 3 MAX. ZEROES = 3
  11. 11. α + β = - coefficient of x Coefficient of x2 = - b a αβ = constant term Coefficient of x2 = c a
  12. 12. α + β + γ = -Coefficient of x2 = -b Coefficient of x3 a αβ + βγ + γα = Coefficient of x = c Coefficient of x3 a αβγ = - Constant term = d Coefficient of x3 a
  13. 13. DIVISION ALGORITHM
  14. 14. •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.

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