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Unformatted text preview: ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ To find the number of factors of a given number, express the number as a product of powers of prime numbers. In this case, 48 can be written as 16 * 3 = (2 4 * 3) Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1) Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ The sum of first n natural numbers = n (n+1)/2 The sum of squares of first n natural numbers is n (n+1)(2n+1)/6 The sum of first n even numbers= n (n+1) The sum of first n odd numbers= n^2 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ To find the squares of numbers near numbers of which squares are known To find 41^2 , Add 40+41 to 1600 =1681 To find 59^2 , Subtract 60^2-(60+59) =3481 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then. eg: x^4+3x^2+2x+6=0 has no positive roots . ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) . Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ For a cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the product of the roots taken two at a time = c/a product of the roots = -d/a ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = - b/a sum of the product of the roots taken three at a time = c/a sum of the product of the roots taken two at a time = -d/a product of the roots = e/a +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) . +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |x| + |y| >= |x+y| (|| stands for absolute value or modulus )...
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This note was uploaded on 01/27/2010 for the course EE 100 taught by Professor Abc during the Spring '10 term at Punjab Engineering College.
- Spring '10