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Mathematics Dictionary
Dr. K. G. Shih
Factor Theory


  • Q01 | - Factor theory
  • Q02 | - Remainder theory
  • Q03 | - Synthetic division
  • Q04 | - F(x) = x^5 -3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12, find F(-1)
  • Q05 | - F(x) = x^5 -3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12, Express in factors

    • Q01. Factor theory

    Factor thoery of functions F(x)
    • If F(a) = 0, then (x - a) is a factor of F(x)
    • Hence F(x) = (x - a)*G(x)
    • If F(-a) = 0, then (x + a) is a factor of F(x)
    • Hence F(x) = (x + a)*G(x)
    Example : Prove that x = 2 is a factor of F(x) = x^2 - 6*x + 8
    • F(2) = 2^2 - 6*2 + 8 = 0
    • Hence (x - 2) is a factor of F(x)
    • Hence F(x) = (x - 2)*(x - 4)

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    Q02. Remainder theory

    Definition
    • If F(x) is divided by (x - a), then remainder is F(a)
    • If F(x) is divided by (x + a), then remainder is F(-a)
    Example : F(x) = x^2 - 6*x + 8 is divided by (x-3), find remainder
    • F(3) = 3^2 - 6*3 + 8 = -1.
    • F(x) divide by (x-3), the remainder is -1.

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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)

    Go to Begin
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