Polynomials(10th) Simplified
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This document discusses polynomials and their properties. It defines polynomials as algebraic expressions involving variables and real numbers. It describes the different types of polynomials based on the number of terms they contain such as monomials, binomials, and trinomials. The document also discusses the degree of a polynomial as well as the maximum number of real zeros a polynomial can have based on its degree. Additionally, it provides relationships between the zeros and coefficients of polynomials. Finally, it introduces the division algorithm for polynomials and how it can be used to verify if one polynomial is a factor of another.
Polynomials(10th) Simplified
- 1. MADE BY:- MUHAMMAD SAJEEL KHAN CLASS:- 10th ‘E’ GIVEN BY:- MR. SANDESH SIR
- 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
- 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. 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.
- 5. 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.
- 6. 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.
- 7. 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.
- 8. 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 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)”. 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
- 9. An nth degree polynomial can have at most “n” real zeroes. Number of real zeroes of a polynomial is less than or equal to degree of the polynomial.
- 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 Note:- “A” and “B” are the zeroes.
- 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 Note:- “A”, “B” and “C” are the zeroes.
- 17. 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)
- 18. 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. QUESTION TYPES!