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**1. Homework Statement**

I am currently working through a chapter on Polynomial Remainder and Factor Theorems in my book, Singapore College Math, Syllabus C.

There were a few problems which I got stuck on:

**25)**The positive or zero integer ##r## is the remainder when the positive integer, ##n## is divided by the positive integer ##p##. Show that ##p## is an exact divisor of ##(n^p-n) - (r^p-r)##. Factorize ##(r^p-r)## in each of the special cases, ##p = 3, 5 , 7,## and deduce that in these cases ##p## is an exact divisor of ##n^p-n##.

**19)**Show that ##a+b## is a factor of ##abc-(a+b+c)(bc+ca+ab)## and factorize this expression completely.

**Example)**Prove that ##(a+b)## is a factor of ##a^2(b+c)+b^2(c+a)+c^2(a+b)+2abc##

and write down the other two factors.

**Proof**

Let ##f(x) = x^2(b+c) + b^2(c+x)+c^2(a+b)+2xbc##. When f(x) is divided by x+b, the remainder is: ... by remainder theorem... 0.

So it's a factor. Now the part I have a problem with:

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since ##(a+b)## is a factor of ##f(x)##

∴ ##f(x) = (a+b)Q(x)##, Q(x) = quotient.

Substituting ##x = a##, we have

##a^2(b+c) + b^2(c+a) + c^2(a+b) + 2abc = (a+b)Q(a)##

The other factors are ##b+c## and ##c+a##

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How do they know that (b+c) and (c+a) are factors?

I can see them in the equation, but how does it work?

**2. Homework Equations**

Factor/remainder theorems and long polynomial division (that's what the chapter was on)

**3. The Attempt at a Solution**

**25)**- This one I have no idea on how to tackle. ##(n^p-n)/p)## is obvious, but that I think is not part of the solution.

**19)**- The second one - I don't know how to do this either :(

**Example)**I understand everything here to the last step - getting the other factors. It didn't show how it was done. It just gave the conclusion. Were the factors found by long division, or were they taken out of the equation because of some rule?