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Factor Theorem Question

  1. Mar 29, 2015 #1
    1. The problem statement, all variables and given/known data
    If the polynomial P(x) = x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find a, b

    2. Relevant equations


    3. The attempt at a solution
    Let (px+q) be a factor of P(x),
    p can possibly be 1 and so can q, according to factor theorem,
    Hence, factors (x+1) or (x-1)
    P(1) = 0, substituting I got -2 as a and P(-1) =0 , I got 2 as a
    If either (x+1) or (x-1) is a factor of P(x), it has to be a factor of T(x),
    T(1) = 0, I got 14 as b in both cases.
    But the correct answer is a=3,-3 and b=-6,6 respectively.
     
  2. jcsd
  3. Mar 29, 2015 #2

    SammyS

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    Hello Rithikha. Welcome to PF !

    You want ##\ p\, x +q \ ## to be a factor of ##\ T(x)\ ## rather than ##\ P(x)\ ##.
     
  4. Mar 29, 2015 #3
    But the constant term is b in T(x), we can get q from it. Or can we?
     
  5. Mar 29, 2015 #4

    HallsofIvy

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    Your error is in assuming that P(x) has a linear factor with rational coefficients (which you do when you use the rational root theorem).

    Instead let y= P(x) and rewrite T(x) in terms of y.
     
  6. Mar 29, 2015 #5

    Ray Vickson

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    You want ##2x^3-16x + b = (d x + c)(x^2 + ax + 1)##. It is easy to see that we must have ##d = 2##, so we need ##2x^3 - 16 x + b = (2x + c)(x^2 + ax +1)##.
     
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