I went through basic algebra literally like 20+ years ago, I generally understand it well, but vaguely remember specifics if any.
Right now I am doing an advanced programming course and one question had an answer which did some basic algebraic simplification. I do not understand how this works anymore. Here is the steps of simplification it went through:
- Step $1:$ $a \cdot b + a \cdot c - (b + c)$
- Step $2:$ $a \cdot (b + c) - (b + c)$
- Step $3:$ $(a - 1) \cdot (b + c)$
I understand step 1 to step 2 fairly well. But how does step 2 end up as step 3?
Could someone point me to a video, tutorial or something that would refresh my understanding of whatever it is that allowed step 2 to get to step 3?
The idea between steps two and three is to factor out $(b+c)$.
This is easier for you to see if we replace $(b+c)$ by a single variable - let's have $z = b+c$. Then, starting at step two, we see
$$a\cdot (b+c) - (b+c) = a\cdot z - z$$
To be sure the next step is perfectly clear, don't forget that $z = 1\cdot z$, so we can say
$$a\cdot z - z = a\cdot z - 1\cdot z$$
Then you can factor out the $z$:
$$a\cdot z - 1\cdot z = z \cdot (a-1)$$
Then you replace $z$ back by $b+c$ again:
$$z \cdot (a-1) = (b+c) \cdot (a-1)$$
To get the exact representation in step four, you can just use the fact that multiplication commutes, i.e. $x\cdot y = y\cdot x$ (order doesn't matter). Thus,
$$(b+c) \cdot (a-1) = (a-1)\cdot (b+c)$$
Of course you can just leave it as $(b+c) \cdot (a-1)$, it's literally the same thing. Purely up to personal taste.