Factoring/Expansion explantion

Question

Sorry if I call something by the wrong name since I didnt learn math in english. ok so for example this: (a+b)(a-b) if you break it down to the second "()" you will end up with this: a+-b could somebody explain why + shows up?

Answer 1

The method to use is called factorisation:

To clear things up, let's give the variables a color: $(\color{blue}{a}\color{red}{+b})(\color{blue}{a}\color{red}{-b})$. We need to apply the following rule:

$\color{blue}{a}$ is multiplied first with $\color{blue}{a}$ and then with $\color{red}{-b}$ and we get the following: $\color{blue}{aa}$ and $\color{red}{-}\color{blue}{a}\color{red}{b}$. Then we multiply $\color{red}{+b}$ with $\color{blue}{a}$ and then $\color{red}{-b}$, we get the following: $\color{red}{b}\color{blue}{a}$ and $\color{red}{-bb}$. Adding these up will get you:

$(\color{blue}{aa})+(\color{red}{-}\color{blue}{a}\color{red}{b})+(\color{red}{b}\color{blue}{a})+(\color{red}{-bb})$. Since $x+-y = x-y$, we get $\color{blue}{aa}\color{red}{-}\color{blue}{a}\color{red}{b}+\color{red}{b}\color{blue}{a}\color{red}{-bb}$.

Also $x\cdot x=x^2$ and $p\cdot q=q\cdot p$, we get $a^2-ab+ab-b^2 = \color{blue}{a^2}-\color{red}{b^2}$