# Algebra question

I’m trying to help my daughter understand algebra. It is sad that she is only in seventh grade and I am have trouble explaining a factoring concept. She is factoring polynomials completely and neither understand how to factor a^4-b^4. I know it has to be simple, and (a-b)^4 certainly is. Any advice?

(a^2 + b^2) (a^2 - b^2). Time to revise “Difference Of Two Squares”!

What would you do if you replaced the [sup]4[/sup] with [sup]2[/sup]?

Since the powers are even you should be able to extrapolate from there.

You can also check out www.mathforum.org. I found that site when I was looking up an equation. It has a lot of cool stuff, including an excellent refresher on probability. Now you can stay one step ahead.

Well, that was ridiculously easy. The second set of problems I am having trouble understanding and therefore explaining is clearly the difference of squares in a different form. Examples are (a+b)^2-(a-c)^2 or a(a^2-9)-2(a+3)^2. The answers I get are just not correct. What process do you use to do this. Is there a different website where such questions are more acceptable?

Well, part of it is just a matter of looking for the appropriate patterns. For example, in this one, the difference of two squares appears in the bolded part:

a**(a^2-9)**-2(a+3)^2

and this becomes:

a**(a-3)(a+3)** - 2(a+3)^2

Noticing that there’s the factor (a+3) in both terms (each side of the subtraction), you can pull that out, and get:

a(a-3) - 2(a+3)

Now expand the part in the brackets and you’ll have a standard-looking polynomial to factor. Whole thing will come out to three roots.

Thanks, but still can’t get (a+b)^2-(a-c)^2. Anyone want to help me out and make me feel dumb?

It’s still a difference of squares: (a + b)[sup]2[/sup] - (a - c)[sup]2[/sup] = ((a + b) - (a - c))((a + b) + (a - c)). I’ll let you simplify.

Most teachers I’ve had would give half credit for this. Now you have another difference of squares, so you get:

a^4 - b^4
(a^2 + b^2)(a^2 - b^2)
(a^2 + b^2)(a + b)(a - b)

Has she learned about i yet? I only ask because apparently my 7th grade algebra learned about it on accident, as the result of a too-curious 11-year-old.

If so, I’d like to be the first to chime in that
(a[sup]2[/sup]+b[sup]2[/sup])=(a+bi)(a-bi)

a[sup]2[/sup] + b[sup]2[/sup] is only factorable if you’re asked to factor over C. If you’re asked to factor over R (which most 7th-grade math homework would do), the correct answer is that a[sup]2[/sup] + b[sup]2[/sup] is irreducible (i.e., not factorable).

Now a[sup]4[/sup] + b[sup]4[/sup], on the other hand…

Most teachers I’ve had would give you part of the answer and let you finish off

Yup. (a^2 + sqrt(ab) + b^2)(a^2 - sqrt(ab) + b^2). And hence with a little application we can factor a^8 - b^8

First of all, if you think those two are the same, you’re drastically mistaken (unless you’re working over a field of characteristic 2).

Anyhow, this looks like a classic example of “difference of squares”. A lot of high school algebra techniques are special tricks and you really don’t want to know the deep ideas. Hell, I don’t really fully grok the deep ideas. In general, this trick is that (x+y)(x-y) = x[sup]2[/sup]-y[sup]2[/sup]. Multiply it out to check.

Apply it first to x = a[sup]2[/sup] and y = b[sup]2[/sup]. Then look for other differences of squares in the result.