Basic question on spivak's calculus

[chemistry1]

http://postimg.org/image/lh7ga876t/

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 what's happening, just curious if there's any justification to it.

Thank you !

 

[SammyS]

http://postimg.org/image/lh7ga876t/

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 what's happening, just curious if there's any justification to it.

Thank you !

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) \$ .

 

[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!

 

[SammyS]

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 .

 

[chemistry1]

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