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  • Math Common Sense 数学小常识

    The sum of arithmetic sequence

    The sum of geometric sequence

    A special formula : n·n! = (n+1)! - n!

    Divisible Test

    4: The last 2 digits are divisible by 4

    6: Divisible by both 2 and 3

    7: Truncate the final digit. Use the rest minus the final digit truncated twice. Check the answer if can be divisible by 7. [ truncation, double, distraction, check ]

    8: The last 3 digits are divisible by 8

    9: Sum of digits is divisible by 9

    11: Add and subtract digits alternating from left to right. Check the answer if can be divisible by 11

      913 ( 9 - 1 + 3 = 11 yes )

      1364 ( 1 - 3 + 6 - 4 = 0 yes )

      987 ( 9 - 8 + 7 = 8 no )

      3729 ( 3 - 7 + 2 - 9 = -11 )

    The number of divisors 

    72 = 23 x 32

    ( 3 + 1 ) x ( 2 + 1 ) = 12

    Each exponent plus one and multiply together

    The factor of polynomial

    a1x+ a2xn-1 + ······ + anx + k = 0

    if it can divide by ( x - s ) , then s is a factor of k/a1

    a3 - 8a + 8 = 0

    8/1 = 8 → ±1, ±2, ±4, ±8

    Vieta's theorem

    ax3 + bx2 + cx + d = 0

    x1 + x2 + x3 = -b/a

    x1x2 + x1x3 + x2x3 = c/a

    x1x2x3 = -d/a

      

    Entire radical : the coefficient is one 

    Mixed radical : 

     

    f(x) ÷ (x-a) = quotient  ······ f(a)

    if f(a) = 0, f(x) is divisible by (x-a)

    Ex.1. What's the remainder of (5x3 + 4x- 12x + 1) ÷ (x - 3)

    5·33+4·32-12·3+1=136

    Ex.2. What's the remainder of (x4+x) ÷ (x2-3x+2)

    let ax+b be the remainder ( the degree of remainder is less 1 than divisor's )

    Solve the divisor, x=1,2

    (14+1) = a·1+b

    (24+2) = a·2 +b

    → a=16, b=-14

    therefore the remainder is 16x-14

    Theorem about Secant, Tangent and Arc

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  • 原文地址:https://www.cnblogs.com/hadilo/p/7865206.html
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