Polynomials

**leo** · August 19th 2009, 05:17 AM

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.

**Moo** · August 19th 2009, 05:26 AM

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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