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I have started the problem set for Chapter one (basic properties of numbers) in Spivak's Calculus (self study). I think I am doing these right, but I have some questions.

As a solid example, problem 1-(iv) says to prove the following:

[tex]x^3 - y^3 = (x-y)(x^2+xy+y^2)\qquad(1)[/tex]

My approach was to add and subtract terms to the left hand side of (1) until it could be factored into the desired form:

[tex]\begin{array}{l}

x^3 - y^3 &= x^3 - y^3 + (x^2y - x^2y) + (xy^2 - xy^2) \\

&= x^3 +x^2y+xy^2 - y^3 - x^2y - xy^2 \\

&=x(x^2+xy+y^2)-y(x^2+xy+y^2) \\

&= (x-y)(x^2+xy+y^2)

\end{array}

[/tex]

Now this seems correct to me, but I feel a littleguiltybecause I only knew what to add and subtract to the LHS of (1) because I knew what I was trying to achieve- the RHS of (1).

My second question is similar, but requires some clarifying of Spivak's intent. In problem 1-(vi) he says: Prove the following:

[tex]x^3 + y^3 = (x+y)(x^2-xy+y^2)\qquad(2)[/tex]

Then he says, "There is a particularly easy way to do this, using (iv), and it will show you how to find a factorization of [itex]x^n+y^n[/itex] whenever n is odd."

Well, I solved this one the exact same way I did with (iv) above: I used the right hand side of (2) to infer what terms to add/subtract and obtained the solution. However, the fact that I do not see what he means by the quoted text above leads me to belive that there was some other approach. That is, I have not discovered a way to factorize [itex]x^n+y^n[/itex] whenever n is odd in my procedure.

Any thoughts are appreciated

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# Spivak: Is his how to approach these proofs?

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