# Thread: real quick question about factoring

1. ## real quick question about factoring

I have this question in my book that reads:

ax - bx + by - ay

it shows me the entire process, even the answer but one part has me confused and is not explained.

x(a-b)+ y(-1)(a-b)
next to it, it says "Factoring out -1 to reverse b-a"

Im just really confused as to where they got that -1 from. It just seems like they pulled it out of thin air.

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heres the entire process up until that point:

ax - bx + by - ay = (ax - bx) + (by-ay)
=x(a-b)+y(b-a)
=x(a-b)+ y(-1)(a-b)

2. If you have the expression $b - a$ you can factor out -1 by dividing both terms by -1, the terms being $b$ and $-a$. $\frac{b}{-1} = -b$, and $\frac{-a}{-1} = a$, so you end up with $-1(-b + a)$, or $-1(a - b)$. Make sense? Basically you can pull out whatever number you want, as long as you divide accordingly. I could factor 543 out of $b - a$, and I would end up with $543 (\frac{1}{543}b - \frac{1}{543}a)$--multiply that out if you don't believe that it's the same thing as $b - a$. In this case they're pulling out a -1.

3. $b-a = -(a-b) = -1 \cdot (a-b)$

4. Originally Posted by NecroWinter
I have this question in my book that reads:

ax - bx + by - ay

it shows me the entire process, even the answer but one part has me confused and is not explained.

x(a-b)+ y(-1)(a-b)
next to it, it says "Factoring out -1 to reverse b-a"

Im just really confused as to where they got that -1 from. It just seems like they pulled it out of thin air.

--------------------------

heres the entire process up until that point:

ax - bx + by - ay = (ax - bx) + (by-ay)
=x(a-b)+y(b-a)
=x(a-b)+ y(-1)(a-b)

when you factor a term from an expression, you actually divide each term in the expression by the factored-out term in order to factor it out. for example:

Say you wanted to factor $\displaystyle x^3 + x$. Of course you would factor out the common $\displaystyle x$ to get $\displaystyle x(x^2 + 1)$

But where did the $\displaystyle x^2$ and the $\displaystyle +1$ come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an $\displaystyle x$, so divide the $\displaystyle x^3$ by $\displaystyle x$, and that's where your $\displaystyle x^2$ comes from. Your 1 comes from dividing $\displaystyle x$ by $\displaystyle x$.

So in other words, what you did was $\displaystyle x \left( \frac {x^3}x + \frac xx \right) = x(x^2 + 1)$

(you can also think of this as multiplying and dividing by x at the same time. this way, you're multiplying by 1 and so you're not changing the value of your expression, just how it looks)

Similarly, $\displaystyle b - a = -1 \left( \frac b{-1} - \frac a{-1} \right) = -(a - b)$.

We usually don't think of factoring in this complicated way, but that's what we're doing when we factor something. as for factoring out -1, you just think of it as factoring out a negative sign, when you do this, you change all the signs of the terms you factored the -1 from. so b - a = -(a - b), the +b became -b and the -a became +a when you factor out a sign. If you multiply out, you get the original expression, which is a good way to check yourself.

Capice? I don't know. I feel like I'm being way too complicated with everything today... maybe I shouldn't be explaining stuff to people

5. Originally Posted by Jhevon

when you factor a term from an expression, you actually divide each term in the expression by the factored-out term in order to factor it out. for example:

Say you wanted to factor $\displaystyle x^3 + x$. Of course you would factor out the common $\displaystyle x$ to get $\displaystyle x(x^2 + 1)$

But where did the $\displaystyle x^2$ and the $\displaystyle +1$ come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an $\displaystyle x$, so divide the $\displaystyle x^3$ by $\displaystyle x$, and that's where your $\displaystyle x^2$ comes from. Your 1 comes from dividing $\displaystyle x$ by $\displaystyle x$.

So in other words, what you did was $\displaystyle x \left( \frac {x^3}x + \frac xx \right) = x(x^2 + 1)$

(you can also think of this as multiplying and dividing by x at the same time. this way, you're multiplying by 1 and so you're not changing the value of your expression, just how it looks)

Similarly, $\displaystyle b - a = -1 \left( \frac b{-1} - \frac a{-1} \right) = -(a - b)$.

We usually don't think of factoring in this complicated way, but that's what we're doing when we factor something. as for factoring out -1, you just think of it as factoring out a negative sign, when you do this, you change all the signs of the terms you factored the -1 from. so b - a = -(a - b), the +b became -b and the -a became +a when you factor out a sign. If you multiply out, you get the original expression, which is a good way to check yourself.

Capice? I don't know. I feel like I'm being way too complicated with everything today... maybe I shouldn't be explaining stuff to people
No, that was actually a fantastic explaination. You dont give yourself enough credit! :P

I like in depth because my problem with math and algebra has always been not understanding why, and you helped me understand why in this specific situation. Thanks!