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9+&+10+English+_+Class+09+CBSE+2020+_Formula+Cheat+Sheet+_+Number+System+&+Polynomials+-+_+11th+Mar.pdf

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9+&+10+English+_+Class+09+CBSE+2020+_Formula+Cheat+Sheet+_+Number+System+&+Polynomials+-+_+11th+Mar.pdf

  1. 1. NUMBER SYSTEM Sl No Type of Numbers Description 1 Natural Numbers N = { 1, 2, 3, 4, 5, . . .} It is the counting numbers 2 Whole Numbers W= { 0, 1, 2, 3, 4, 5, . . .} It is the counting numbers + zero 3 Integers Z = {. . . -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 . . .} 4 Positive Integers Z+ = { 1, 2, 3, 4, 5, . . . } 5 Negative integers Z– = {. . . -4, -3, -2, -1 }
  2. 2. Sl No Type of Numbers Description 6 Rational Numbers A number is called rational if it can be expressed in the form p/q where p and q are integers (q>0). Ex: 4/5 7 Irrational Numbers A number is called rational if it cannot be expressed in the form p/q where p and q are integers (q> 0). Ex: √2, Pi, … etc 8 Real Numbers A real number is a number that can be found on the number line. All rational and irrational numbers makes the collection of Real Numbers. [Denoted by the letter R]
  3. 3. Natural Numbers N 1, 2, 3, . . . Whole Numbers W 0, 1, 2, 3, . . . Integers Z . . ., -2, -1, 0, 1, 2, 3, . . . Rational Numbers Q 0 1 5 ½ -⅔ -9 Irrationals √2 √3 𝝅 0.1011011 1011110... REAL NUMBERS Real Numbers
  4. 4. Sl No Type of Numbers Description 9 Real numbers & their decimal Expansions The decimal expansion of a rational number is either terminating or non terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. The decimal expansion of an irrational number is non-terminating non- recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.
  5. 5. Sl No Type of Numbers Description 10 Operations on Real numbers The sum or difference of a rational number and an irrational number is irrational The product or quotient of a non-zero rational number with an irrational number is irrational. If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.
  6. 6. Sl No Type of Numbers Description 11 Rationaliza tion Rationalizing a denominator is a technique to eliminate the radical from the denominator of a fraction. Rationalizing the denominator helps understand the quantity better and is helpful to plot them on the numberline.
  7. 7. Sl No Type of Numbers Description 12 Laws of Exponents
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  17. 17. Sl no Terms Description 1 Definition A polynomial expression P(x) in one variable is an algebraic expression in x where power of the variable is whole number and coefficients are real numbers. Polynomials
  18. 18. Sl no Terms Description 3 Degree Highest power of the variable in a polynomial is the degree of the polynomial. 4 Terms of a polynomial expression The several parts of a polynomial separated by ‘+’ or ‘-‘ operations are called the terms of the expression. Ex :
  19. 19. Type of polynomial Degree Form Constant 0 P(x) = a Linear 1 P(x) = ax + b Quadratic 2 P(x) = ax2 + ax + b Cubic 3 P(x) = ax3 + ax2 + ax + b Bi-quadratic 4 P(x) = ax4 + ax3 + ax2 + ax + b #5 Types of Polynomial based on their Degrees
  20. 20. Type of polynomial Degree Form Monomial 1 Polynomials having only one term are called monomials (‘mono’ means ‘one’). e.g., 13x2 Binomial 2 Polynomials having only two terms are called binomials (‘bi’ means ‘two’). e.g., (y30 + √2) Trinomial 3 Polynomials having only three terms are called trinomials (‘tri’ means ‘three’). e.g., (x4 + x3 + √2), (µ43 + µ7 + µ) and (8y – 5xy + 9xy2) are all trinomials #6 Types of Polynomial based on the number of terms
  21. 21. Sl no Terms Description 7 Zeroes or Roots of a Polynomia l A zero of a polynomial p(x) is a number c such that p(c) = 0. If P(a) = 0, then ‘a’ is the zero of the polynomial P(x) and the root of the polynomial equation P(x) = 0. Note: ● A non-zero constant polynomial has no zero. ● By convention, every real number is a zero of the zero polynomial. ● The maximum number of zeroes of a polynomial is equal to its degree.
  22. 22. Sl no Terms Description 8 Factor Theorem When a polynomial f (x) is divided by (x – a), the remainder = f (a). And, if the remainder f (a) = 0, then (x – a) is a factor of the polynomial f(x). Note: We have to know factor theorem in order to factorize cubic polynomials.
  23. 23. Sl no Terms Description 9 Factorization of a Polynomial By taking out the common factor: If we have to factorize x2 –x then we can do it by taking x common. x(x – 1) so that x and x-1 are the factors of x2 – x. By grouping: ab + bc + ax + cx = (ab + bc) + (ax + cx) = b(a + c) + x(a + c) = (a + c)(b + x) By splitting the middle term: Write the given quadratic Polynomial in standard form. Find the Product of a and c. List down all factors of ac in pairs. Select a pair of factors such that their sum is ‘b’. Now split the middle term ‘b’, in terms of the factors found.
  24. 24. Sl no Terms Description 10 Algebraic Identities
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