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Math Help - Polynomials

  1. #1
    leo
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    Arrow Polynomials

    Hello, can anyone help me with this question? Thanks in advance

    Find the remainder when (x^100)+5 is divided by (x^2)-2x+1.
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  2. #2
    Moo
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    Hello,

    The remainder has to be a polynomial of at most degree 1.
    This means that it is in the form R(x)=ax+b, where a and b can be equal to 0.

    Let's denote P(x)=x^{100}+5 and note that x^2-2x+1=(x-1)^2

    What we'll do is studying the zeroes of the polynomials...

    We have P(x)=(x-1)^2Q(x)+R(x)

    where Q is the quotient.


    If we let x=1, we have P(1)=6=0*Q(1)+R(1)
    Thus \boxed{R(1)=a+b=6}

    If we differentiate, we have P'(x)={\color{red}2(x-1)Q(x)+(x-1)^2Q(x)}+R'(x)

    But P'(x)=100x^{99} and R'(x)=a

    Again, if we let x=1, the red part will be equal to 0.

    Which gives \boxed{1=a}

    Thus b=... and the remainder R is...
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