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Factor and Remainder Thoerem

  1. Jan 22, 2008 #1
    A couple of hard questions about the Factor and Remainder Thoerem that I'm having a hard time with.

    1. The problem statement, all variables and given/known data

    18) f(x) = 2x3 + x2 – 5x + c, where c is a constant.
    Given that f(1) = 0,

    (a) find the value of c

    (b) factorise f(x) completely,

    (c) find the remainder when f(x) is divided by (2x – 3).

    19) f(x) = x3 – 2x2 + ax + b, where a and b are constants.

    When f(x) is divided by (x – 2), the remainder is 1.

    When f(x) is divided by (x + 1), the remainder is 28.

    (a) Find the value of a and the value of b.

    (b) Show that (x – 3) is a factor of f(x).

    2. Relevant equations


    3. The attempt at a solution

    These are the last two questions of the homework, and the only ones I am having some difficulty with. The other 17 questions I have finished and am happy with. I reckon I can do 18) a) though;

    f(1) = 2(1)^3 + 1² - 5(1) + c
    = 2 + 1 -5 + c
    c = 2

    Right? Any help with the other questions is very welcome! Thanks for your time.
  2. jcsd
  3. Jan 22, 2008 #2


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    Homework Helper

    Yes c=2 making [itex]f(x)=2x^3 + x^2-5x+2[/itex]

    The remainder and factor theorem states that if f(x) is any polynomial and f(x) is divided by x-a then the remainder is f(a).If f(a)=0 then (x-a) is a factor of f(x).

    From the theorem above.

    f(1)=0. This means that (x-1) is a factor of f(x). Now you can just do long division or synthetic division and you will get the other quadratic factor. which you can then factorize further if possible.
  4. Jan 23, 2008 #3
    Hey thanks, that's a great explanation, I've finished off question 18 quite comfortably. Any help with question 19?
  5. Jan 23, 2008 #4
    They are leading you to a point where you will have two equations in two unknowns.

    By the remainder theorem, the first part says that f(x) = (x-2)q(x) + f(2), where q(x) is a quadratic polynomial. This tells you something about f(2). The second part says something similar. Can you put the rest together?
    This should be accomplished using the two facts above in combination with the remainder theorem.
    Once you know a and b you should be able to prove this - remember what it means about f(3).
    Last edited: Jan 23, 2008
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