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Mathematics Dictionary
Dr. K. G. Shih
False Proof


  • Q01 | - Prove that 2 = 0
  • Q02 | - Prove that 2 = 1
  • Q03 | -
  • Q04 | -
  • Q05 | -

    • Q01. Prove that 2 = 0

    Proof
    • Let a = 1 and b = 1
    • Hence a = b
    • Hence a^2 = b^2
    • Hence a^2 - b^2 = 0
    • Hence (a + b)*(a - b) = 0
    • Divide both sides by (a - b)
    • Hence a + b = 0
    • Hence 1 + 1 = 0
    Note
    • Since a - b = 0 and it can not be used as divisior

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    Q02. Prove that 2 = 1

    Proof
    • Let a = 1 and b = 1
    • a = b
    • Both sides times a
    • a^2 = a*b
    • Both sides minus b^2
    • a^2 - b^2 = a*b - b^2
    • Hence (a + b)*(a - b) = b*(a - b)
    • Divide both sides by (a - b)
    • Hence a + b = b
    • Hence 1 + 1 = 1
    • Hence 2 = 1
    Note
    • Since a - b = 0 and it can not be used as divisior

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    Q03. Synthetic division

    Prove that (x^3 + 1)/(x + 1) = x^2 - x + 1
    • Put coefficients of x^3 + 1 in a row
    • For missing terms the coefficients put a zero
    • Step 1
      • ..... 1 +0 +0 +1 | -1 (For x + 1)
      • ....... -1 (1 times -1 = -1)
      • ..... ------------
      • ..... 1 -1 (0 - 1 = -1)
    • Step 2
      • ..... 1 +0 +0 +1 | -1 (For x + 1)
      • ....... -1 +1 (-1 times -1 = +1)
      • ..... ------------
      • ..... 1 -1 +1 (0 + 1 = +1)
    • Step 3
      • ..... 1 +0 +0 +1 | -1 (For x + 1)
      • ....... -1 +1 -1 (+1 times -1 = -1)
      • ..... ------------
      • ..... 1 -1 +1 +0 (+1 - 1 = +0)
    • Since remainder is zero
    • Hence x^3 + 1 = (x + 1)*(x^2 - x + 1)
    Prove that (x^3 - 1)/(x - 1) = x^2 + x + 1
    • Put coefficients of x^3 - 1 in a row
    • For missing terms the coefficients put a zero
    • ..... 1 +0 +0 -1 | 1 (Note 1)
    • ....... +1 +1 +1 |
    • ..... ------------
    • ..... 1 +1 +1 +0 (Note 2)
    • Hence x^3 - 1 = (x - 1)*(x^2 + x + 1)
    Notes
    • Note 1 : 1 stands for (x - 1)
    • Note 2 :
      • 2nd number = +0 + 1
      • 3rd number = +0 + 1
      • 4th number = -1 + 1 = 0 (remainder is zero)
    Example : Express x^3 - 6*(x^2) + 11*x - 6 in factor form
    • We have to try (x - 1), (x - 2), (x - 3), (x + 1), (x + 2), ....
    • That is all factors of last term number 6
    • They are -1, -2, -3, -6, 1, 2, 3, 6
    • We try (x - 1)
      • ..... 1 -06 +11 -06 | +1 (For x - 1 we use +1)
      • ....... +01 -05 +06 |
      • ..... ------------
      • ..... 1 -05 +06 +00
    • Since remainder is zero
    • Hence (x - 1) is a factor
    • We have x^3 - 6*(x^2) + 11*x - 6 = (x - 1)*(x^2 - 5*x + 6)
    • Since x^2 - 5*x + 6 = (x - 2)*(x - 3)
    • Hence x^3 - 6*(x^2) + 11*x - 6 = (x - 1)*(x - 2)*(x - 3)
    Note
    • If coefficient of first term is 2 and last term is 6
    • we have to try 1/2, 3/2, ....

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    Q4. F(x) = x^5 -3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12

    Find F(-1)
    • Step 1
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 (1 times -1)
      • ...... ------------------------
      • ...... +1 +02 (3 - 1 = 2)
    • Step 2
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 -02 (2 times -1)
      • ...... ------------------------
      • ...... +1 +02 -07 (-5 - 2 = -7)
    • Step 3
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 -02 +07 (-7 times -1)
      • ...... ------------------------
      • ...... +1 +02 -07 -08 (-15 + 7 = -8)
    • Step 4
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 -02 +07 +08 (-8 times -1)
      • ...... ------------------------
      • ...... +1 +02 -07 -08 +12 (4 + 8 = 12)
    • Step 5
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 -02 +07 +08 -12 (12 times -1)
      • ...... ------------------------
      • ...... +1 +02 -07 -08 +12 -00 (12 - 12 = 0)
    • Hence the remainder is zero
    • That is F(x) is divisible by (x + 1)
    • That is also (x + 1) is a factor of F(x)
    • F(x) = (x + 1)*(x^4 + 2*(x^3) - 7*(x^2) - 8*x + 12)

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    Q5. F(x) = x^5 -3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12

    Express F(x) in product of factors
    • Step 1 : Try (x + 1)
      • ...... +1 +03 -05 -15 +04 +12 | -1
      • ......... -01 -02 +07 +08 -12 (12 times -1)
      • ...... ------------------------
      • ...... +1 +02 -07 -08 +12 -00 (12 - 12 = 0)
    • F(x) = (x + 1)*(x^4 + 2*(x^3) - 7*(x^2) - 8*x + 12)
    • Step 2 : Try (x + 2) is afactor of x^4 + 2*(x^3) - 7*(x^2) - 8*x + 12
      • ...... +1 +02 -07 -08 +12 | -2
      • ......... -02 -00 +14 -12
      • ...... --------------------
      • ...... +1 -00 -07 +06 +00
    • F(x) = (x + 1)*(x + 2)*(x^3 - 7*x + 6)
    • Step 3 : Try (x - 1) is afactor of x^3 - 7*x + 6
      • ...... +1 +00 -07 +06 | +1
      • ......... +01 +01 -06
      • ...... --------------------
      • ...... +1 +01 -06 +00
    • F(x) = (x + 1)*(x + 2)*(x - 1)(x^2 + x - 6)
    • Since (x^2 + x - 6) = (x + 3)*(x - 2)
    • F(x) = (x + 1)*(x + 2)*(x + 3)*(x - 1)*(x - 2)

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