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Math Help - Remainder Theorem

  1. #1
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    Remainder Theorem

    Use the remainder theorem to factorise :

    1) x^3 +3x^2 - 4

    2) x^4 + 3x^3 + 5x^2 + 9x + 6
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  2. #2
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    Quote Originally Posted by Jen1603 View Post
    Use the remainder theorem to factorise :

    1) f(x)=x^3 +3x^2 - 4
    The rational roots theorem tells us if this has rational roots, they will be
    \pm 1, \pm 2, \pm4

    Test each one using the remainder theorem to see which one(s) satisfy f(x)=0

    I found f(-2)=0, so x+2 is a factor.

    To find the remaining factors, you could test the rest of the possible rational roots using the remainder theorem or use synthetic division with divisor -2to reduce the polynomal to a quadratic that you can factor.

    Code:
     
    -2 | 1  3   0  -4
           -2  -2   4
        ------------
         1  1  -2  0
    The quadratic remaining after factoring x+2 is x^2+x-2

    That factors into (x+2)(x-1)

    \boxed{f(x)=(x+2)(x+2)(x-1)}

    Quote Originally Posted by Jen1603 View Post
    2) f(x)= x^4 + 3x^3 + 5x^2 + 9x + 6
    Test these factors \pm 1, \pm 2, \pm 3, \pm 6

    There are no sign changes in this polynomial, so no need to check for positive zero because there aren't any (Descartes Rule of Signs)
    Last edited by masters; November 3rd 2008 at 01:28 PM.
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