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Math Help - Finding remainder of polynomial division

  1. #1
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    Finding remainder of polynomial division

    Find the remainder if the polynomial f(x) = 3x^100 + 5x^87 - 4x^40 + 2x^21 - 6 divided by p(x) = x + 1


    The thing is, I can't use synthetic OR long division BECAUSE I'd have a VERY VERY long equation (since you have to fill in the missing degrees: 99,98,97,96,95......etc).

    How would you solve this?
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  2. #2
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    Quote Originally Posted by mwok View Post
    Find the remainder if the polynomial f(x) = 3x^100 + 5x^87 - 4x^40 + 2x^21 - 6 divided by p(x) = x + 1


    The thing is, I can't use synthetic OR long division BECAUSE I'd have a VERY VERY long equation (since you have to fill in the missing degrees: 99,98,97,96,95......etc).

    How would you solve this?

    Use the Remainder theorem and find f(-1).
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    According to The Remainder Theorem

    How exactly would I use the remainder theorem????
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  4. #4
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    Quote Originally Posted by mwok View Post
    According to The Remainder Theorem

    How exactly would I use the remainder theorem????
    The remainder theorem: When a polynomial P(x) is divided by x - a , the remainder is P(a)!

    Your divisor is (x + 1). This means a = -1. Now, find P(a)...which in your case is P(-1).

    P(-1)=3(-1)^{100} + 5(-1)^{87} - 4(-1)^{40} + 2(-1)^{21} - 6

    This should yield the remainder you seek.
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  5. #5
    Newbie Black Kawairothlite's Avatar
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    you should use it at this way:

    \frac{{3x^{100}+5x^{87}-4x^{40}+2x^{21}-6}}{{x+1}}

    so let's say that:

    p(x)=3x^{100}+5x^{87}-4x^{40}+2x^{21}-6

    and q(x) is the unknown answer

    so rewriting the cocient and according to the remainder theorem:

    p(x)=(x+1)q(x)+r(x) where r(x) is the remainder

    then:

    p(-1)=(0)q(-1)+r(-1)

    so p(-1)=r(-1)

    3(-1)^{100}+5(-1)^{87}-4(-1)^{40}+2(-1)^{21}-6=r(-1)

    knowing the remainder then use synthetic division to do the operation which is easier to do.
    Last edited by Black Kawairothlite; October 29th 2008 at 10:25 AM. Reason: late xD
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  6. #6
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    I finally got it, thanks!
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  7. #7
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    Quote Originally Posted by Black Kawairothlite View Post
    you should use it at this way:

    \frac{{3x^{100}+5x^{87}-4x^{40}+2x^{21}-6}}{{x+1}}

    so let's say that:

    p(x)=3x^{100}+5x^{87}-4x^{40}+2x^{21}-6

    and q(x) is the unknown answer

    so rewriting the cocient and according to the remainder theorem:

    p(x)=(x+1)q(x)+r(x) where r(x) is the remainder

    then:

    p(-1)=(0)q(-1)+r(-1)

    so p(-1)=r(-1)


    3(-1)^{100}+5(-1)^{87}-4(-1)^{40}+2(-1)^{21}-6=r(-1)

    knowing the remainder then use synthetic division to do the operation which is easier to do.
    Book states f(x) = p(x) * q(x) + r(x) r(x) being remainder

    where did you get the zero in (0)q(-1)?
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  8. #8
    Newbie Black Kawairothlite's Avatar
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    when you replace x @ (x+1) to -1 then (-1+1)=(0)

    btw i saw that on the link u put there
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