Published on: **Mar 4, 2016**

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Presentations & Public Speaking

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- 1. A presentation on Presented By;- NAME – Ashish Pradhan , Durgesh Kumar CLASS- X – ‘A’ ROLL NO-27 , 26
- 2. INTRODUCTION GEOMETRICAL MEANING OF ZEROES OF THE POLYNOMIAL RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL DIVISION ALGORITHM FOR POLYNOMIAL
- 3. Polynomials are algebraic expressions that include real numbers and variables. The power of the variables should always be a whole number. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication. Polynomials contain more than one term. Polynomials are the sums of monomials. A monomial has one term: 5y or -8x2 or 3. A binomial has two terms: -3x2 2, or 9y - 2y2 A trinomial has 3 terms: -3x2 2 3x, or 9y - 2y2 y The degree of the term is the exponent of the variable: 3x2 has a degree of 2. When the variable does not have an exponent - always understand that there's a '1' e.g., 1x Example: x2 - 7x - 6 (Each part is a term and x2 is referred to as the leading term)
- 4. WHAT IS A POLYNOMIAL
- 5. Let “x” be a variable and “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 degree is the term with the greatest exponent Recall that for y2, y is the base and 2 is the exponent For example: p(x) = 10x4 + ½ is a polynomial in the variable x of degree 4. p(x) = 8x3 + 7 is a polynomial in the variable x of degree 3. p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the variable x of degree 3. p(x) = 8u5 + u2 – 3/4 is a polynomial in the variable x of degree 5.
- 7. More information of degree
- 8. A real number α is a zero of a polynomial f(x), if f(α) = 0. e.g. f(x) = x³ - 6x² +11x -6 f(2) = 2³ -6 X 2² +11 X 2 – 6 = 0 . Hence 2 is a zero of f(x). The number of zeroes of the polynomial is the degree of the polynomial. Therefore a quadratic polynomial has 2 zeroes and cubic 3 zeroes.
- 9. For example: f(x) = 7, g(x) = -3/2, h(x) = 2 are constant polynomials. The degree of constant polynomials is ZERO. For example: p(x) = 4x – 3, p(y) = 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.
- 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 polynomial is always in the form:- 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) = 5x3 – 2x2 + 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.
- 11. If p(x) is a polynomial and “y” is any real no. then real no. obtained by replacing “x” by “y”in p(x) is called the value of p(x) at x = y and is denoted by “p(y)”. 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. For example:- Value of p(x) at x = 1 p(x) = 2x2 – 3x – 2 p(1) = 2(1)2 – 3 x 1 – 2 = 2 – 3 – 2 = -3 For example:- 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 ZERO OF A POLYNOMIAL
- 12. ☻ A + B = - Coefficient of x Coefficient of x2 = - b a ☻ AB = Constant term Coefficient of x2 = c a Note:- “A” and “B” are the zeroes.
- 13. Number of real zeroes of a polynomial is less than or equal to degree of the polynomial. An nth degree polynomial can have at most “n” real zeroes.
- 14. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 LINEAR FUNCTION DEGREE =1 MAX. ZEROES = 1
- 15. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 QUADRATIC FUNCTION DEGREE = 2 MAX. ZEROES = 2
- 16. Relationship between the zeroes and coefficients of a cubic polynomial • Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx • Then, sum of zeroes(α+β+γ) = -b = -(coefficient of x²) a coefficient of x³ αβ + βγ + αγ = c = coefficient of x a coefficient of x³ Product of zeroes (αβγ) = -d = -(constant term) a coefficient of x³
- 17. GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 CUBIC FUNCTION DEGREE = 3 MAX. ZEROES = 3
- 18. If p(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 : P(x) = q(x) g(x) + r(x), Where r(x) = 0 or degree r(x) < degree g(x)
- 19. QUESTIONS BASED ON POLYNOMIALS I) Find the zeroes of the polynomial x² + 7x + 12and verify the relation between the zeroes and its coefficients. f(x) = x² + 7x + 12 = x² + 4x + 3x + 12 =x(x +4) + 3(x + 4) =(x + 4)(x + 3) Therefore,zeroes of f(x) =x + 4 = 0, x +3 = 0 [ f(x) = 0] x = -4, x = -3 Hence zeroes of f(x) are α = -4 and β = -3.
- 20. 2) Find a quadratic polynomial whose zeroes are 4, 1. sum of zeroes,α + β = 4 +1 = 5 = -b/a product of zeroes, αβ = 4 x 1 = 4 = c/a therefore, a = 1, b = -4, c =1 as, polynomial = ax² + bx +c = 1(x)² + { -4(x)} + 1 = x² - 4x + 1
- 21. THE END