# Math Help - Factoring Problem

1. ## Factoring Problem

Hello all.

I have attached a factoring problem I'm not quite sure on.

Also, does the form of a factorisation allow the addition and subtraction of terms, or must they be bracketed only? For example, the form of (x+y)(x-y) is valid but is something like 2(x+y) + 3(x-z), as an answer to a different possible problem, also valid form?

I've been able to produce something of the second form but I can tell I'm floundering. I've gone off on some wild tangent and it's become too convoluted which is when I can see I need help. A walkthrough of this problem would be most appreciated.

And, as it's my first post here, I'll tell you a little about myself. My math skills are appalling. I sincerely want to become competent in maths and chemistry for two reasons: I want to study medical science, or a similar health science; I've run from maths before and want to know I'm capable of conquering high school mathematics at least.

A rather large problem for me is that I've virtually no math history. I payed no attention in high school and dropped out. I finished through an adult study program two years later and did not complete any science subjects (except for biology). I started late for both of the years I was studying and could not keep pace with the math classes on both occasions. I've been to university since; though I’ve only studied some accounting and creative writing and now want to change direction. It's very important to me that I lift myself out of this hole of mathematical incompetence. The help of the forum members here would be greatly appreciated.

I hope this is the first of many posts and many thanks to anyone with the time to help.

2. Originally Posted by jerry
Hello all.

I have attached a factoring problem I'm not quite sure on.

Also, does the form of a factorisation allow the addition and subtraction of terms, or must they be bracketed only? For example, the form of (x+y)(x-y) is valid but is something like 2(x+y) + 3(x-z), as an answer to a different possible problem, also valid form?
Here is how I did this:

Multiply through first, we get:

$xy^2 - xz^2 + yz^2 - yx^2 + zx^2 - zy^2$

There may be more than one right answer, but I grouped the biggie terms, i.e., the squared terms as follows:

$x^{2}(z - y) + y^{2}(x - z) + z^{2}(y - x)$

For the answer you got, the best way is to multiply through to see what you get. For your answer, you see that you get cubic terms.

$x^3 + xy^2 - yx^2 - y^3 - x^3 - xz^2 - zx^2 - z^3 + y^3 + yz^2 - zy^2 - z^3$

Without simplifying the above, we have a $2z^3$ term. Looking at your original problem, at no time was a z multiplied by a $z^2$

I came up with the same answer as you before coming to what is obviously incorrect. I didn't think it was right though because it just seemed like I'd rearranged it. However, I see now that it's an improvement in that there are less squares etc...

I think, with that problem, I just didn't know how to tell if I had a correct answer. Combine that with self-doubt and you get convolutions like the solution I posted.

Thank you for the help!

4. It's a bit difficult to tell what they are looking for, they may just want you to go like this:
$x(y-z)(y+z) + y(z-x)(z+x) + z(x-y)(x+y)$

This works because:

$a^2 - b^2$

you can factor it like this: add and subtract ab (note that ab - ab = 0 so we are adding the equivalent of zero, and thus not changing the problem)
$= a^2 + ab -ab -b^2$

group them together
$= (a^2 + ab) + (-ab -b^2)$

factor "a" out of the first group, "-b" out of the second group
$= a(a + b) - b(a +b)$

factor (a+b) out of each group
$= (a-b)(a + b)$

For that last step, if it was hard to see, replace "(a+b)" with some other letter, like "c" it would then look like this:
$ac - bc = c(a-b)$

Then you can replace c with "(a+b)" again, and see that it looks proper.

So from this, we can see that any time we have any terms a^2 and b^2 we can directly factor it to this:
$a^2 - b^2 = (a+b)(a-b)$

Be careful, though, if they have a plus sign instead of a minus sign, it does not work.

So for your problem, the first group is
$x(y^2 - z^2)$

$=x(y^2 +yz - yz - z^2)$

$=x(y(y +z) -z(y + z))$

$=x(y -z)(y + z)$

Then you can subsequently do that for the rest of the terms.

...but the instructions are somewhat ambiguous, this may not be what they are looking for. They may be looking for what mathceleb posted, it is difficult to say.