I have been presented with the following question:
Factorise $$a^2-4b^2\;.$$
I know that the answer is $(a-2b)(a+2b)$, but I have no idea how to get there. Could anybody please help me with this?
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@Ed_4434 Think you can remove the comment now... – Simply Beautiful Art Oct 01 '16 at 23:06
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There isn't much to knowing how to get there. Just notice that $$\begin{align}(a-2b)(a+2b) &= a^2 -2ab +2ab -4b^2 \\&= a^2 -4b^2\end{align}$$
In general, the expression you are given is a difference of squares because $$a^2 -4b^2 = a^2 - (2b)^2\;.$$ Any time you have a difference of squares, you can factor it: $$ x^2-y^2 = (x+y)(x-y)\;. $$
Mike Pierce
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I think the OP wants to start with $a^2-4b^2$ and work backwards to derive $(a-2b)(a+2b)$ – Simply Beautiful Art Oct 01 '16 at 22:53
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@SimpleArt Yes this is true, I am needing to see how to get from the question to the answer in a way that is memorable and relatable – Aztec warrior Oct 01 '16 at 22:56
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But to factor it, you really do just have to notice it's a difference of squares. And in general there is no procedural way to factor a polynomial expression; you just have to guess and check. And since I saw it was a difference of squares, I was able to make a good first guess. :) – Mike Pierce Oct 01 '16 at 22:57
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@MikePierce Nah, don't say that mate. You gotta believe in that real MVP algebra that solves quartic polynomials using trigonometric functions. :D – Simply Beautiful Art Oct 01 '16 at 23:01
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I see now that I must simply remember the layout for future questions, and realise it is allowed to change the base accordingly. – Aztec warrior Oct 01 '16 at 23:09
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@FlewittConnor That is definitely the recommended path. $x^2-y^2$ is one of those important simple factors you just have to remember. Also, there's the quadratic formula if you want to get silly :D – Simply Beautiful Art Oct 01 '16 at 23:11
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To start, I have to ask you what the true meaning behind 'factoring' is. To me, it is as follows...
To factor is to derive values for which a polynomial expression is equal to $0$.
So, what we want to do is to set the expression equal to $0$, then solve for $a$ (or $b$).
$$a^2-4b^2=0$$
$$a^2=4b^2$$
$$a=\pm\sqrt{4b^2}=\begin{cases}r_1=\ \ \ 2b\\r_2=-2b\end{cases}$$
Putting this into factored polynomial form:
$$a^2-4b^2=(a-r_1)(a-r_2)=(a-2b)(a+2b)$$
Simply Beautiful Art
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