A little help with algebra (homework) I was brushing up on my algebra a little, and I've seen that I've become rusty with some of the concepts. Indications for solving these would perhaps be better suited rather than full answers, as the goal is to figure it out myself.


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*Show that for any $a, b \in \mathbb{R}$ (without induction, just using algebra) that $a^3 - b^3 = (a - b)(a^2 + 2ab + b^2)$ and something similar but for $a^3 + b^3$.

*Factorize the following: $(3x + 1)^2 - (x+3)^2$ and other such expressions
And if someone could provide a concise and simple reference for the rules of expansions, factorizing and related topic, I would be most appreciative.

The simplest way to verify an identity like $a^3-b^3=(a-b)(a^2+2ab+b^2)$ is to multiply out the righthand side and verify that you do indeed get the lefthand side. If you try it in this case, however, you’ll fail:
$$\begin{align*}
(a-b)(a^2+2ab+b^2)&=a(a^2+2ab+b^2)-b(a^2+2ab+b^2)\\
&=\left(a^3+2a^2b+ab^2\right)-\left(a^2b+2ab^2+b^3\right)\\
&=a^3+2a^2b+ab^2-a^2b-2ab^2-b^3\\
&=a^3-b^3+\left(2a^2b-a^2b\right)+\left(ab^2-2ab^2\right)\\
&=a^3-b^3+a^2b-ab^2\;,
\end{align*}$$
and $a^2b-ab^2=ab(a-b)$ certainly isn’t guaranteed to be zero. (It’s zero if and only if either $a=0,b=0$, or $a=b$.) Thus, in general $a^3-b^3\ne(a-b)(a^2+2ab+b^2)$. The correct identity is $$a^3-b^3=(a-b)(a^2+ab+b^2)\;,$$ as you can check by multiplying out the righthand side: this time everything will cancel out except $a^3-b^3$.
The trick to factorizing an expression like $(3x + 1)^2 - (x+3)^2$ is to recognize that it has the form $a^2-b^2$, where $a=3x+1$ and $b=x+3$, and to recall the basic factorization formula $$a^2-b^2=(a-b)(a+b)\;.$$
A few of the standard basic formulas can be found here, together with a link to a practice page. Here is the start of a set of three pages on the topic, with examples. Googling on factoring formulas, with or without quotes, will turn up many more such resources.
