Basic question on spivak's calculus

1. Dec 23, 2013

chemistry1

http://postimg.org/image/lh7ga876t/ [Broken]

Hi, I have a basic question concerning definition of the word 'factorization'. Does Spivak consider factorization as development of factors ? He goes from saying the "factorization" x2−3x+2=(x−1)(x−2) is really a triple use of P9 and goes on showing development.

P9 says : If a,b, and c are any numbers, then : a⋅(b+c)=a⋅b+a⋅c
Also, when Spivak does the following : (x−1)(x−2)=x(x−2)+(−1)(x−2) does he use any property or just assumes it as like this ? I know whats happening, just curious if there's any justification to it.

Thank you !

Last edited by a moderator: May 6, 2017
2. Dec 23, 2013

SammyS

Staff Emeritus
Note: Use the X2 icon for exponents (superscripts).

Here's the image you posted:

I suppose Spivak does assume that x-1 is the same as x + (-1) .

Then of course, $\displaystyle\ (x-1)(a)\$ is equivalent to $\displaystyle\ x(a)+(-1)(a)\$ . Correct? (Assuming we can distribute from the left as well as from the right.)

Then just let $\displaystyle\ a = (x-2) \$ .

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3. Dec 23, 2013

chemistry1

Yeah, that I understood. The other thing which I don't understand is why does he talk about using P9 to factorize if he's showing the development of factors. How does it make any sense ?thank you!

4. Dec 23, 2013

SammyS

Staff Emeritus
It looks like he's using P9 to expand (multiply out) the factorized form, (x-1)(x-2), verifying that it is the correct factorization for x2 - 3x + 2 .

5. Dec 24, 2013

chemistry1

Yeah, I noticed that. I just was expecting the inverse, the factorization. Anyway, thank you for the help!