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Math Help - Dividing Polynomials

  1. #1
    Senior Member sakonpure6's Avatar
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    Dividing Polynomials

    Hi, I am struggling with the following question >.<
    a) 8x^3 + 10x^2 -px - 5 is divisible by 2x-1. There is no remainder. Find the value of p.

    b) When x^6+x^4-2x^2+k is divided by 1+x^2, the remainder is 5. Find the value of k.
    I tried, but could not come up with anything. Thank you in advance.
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  2. #2
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    Re: Dividing Polynomials

    Quote Originally Posted by sakonpure6 View Post
    Hi, I am struggling with the following question >.<
    I tried, but could not come up with anything. Thank you in advance.
    Do you actually know how to do polynomial "long division"?
    If not then that is where you start.
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  3. #3
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    Re: Dividing Polynomials

    You don't need polynomial division. If 8x^3 + 10x^2 -px - 5 is divisible by 2x-1, then it must have x=\frac{1}{2} as one of its roots. Plug in x=\frac{1}{2} into 8x^3 + 10x^2 -px - 5 = 0 and solve for p. Do you see how this works?

    Similarly, if x^6+x^4-2x^2+k has a remainder of 5 when divided by 1+x^2, then x^6+x^4-2x^2+k - 5 must have all of the roots which 1+x^2 has. Plug in x = i OR x = -i into x^6+x^4-2x^2+k - 5 = 0 and solve for k.
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  4. #4
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    Re: Dividing Polynomials

    Thought you might like to know why:

    if x-r divides p(x), p(x)=(x-r)q(x), and p(r)=0

    if x-r divides p(x) with remander R, p(x)=(x-r)q(x) + R, and p(r)=R
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