Question

Hello everyone. It's been awhile since I've been up here. I have one about a polynomial / another polynomial. Here it is:

(x^2-y^2+2yz-z^2)/(x+y-z)

Any thoughts?

Answer 1

You do long division on polynomials just like on numbers. What is the biggest thing you could multiply (x+y-z) by and have the result be < or = to the numerator? After you subtract the result from the numerator, what is left? Is there another value you could multiply (x+y-z) by that would be less than or equal to that? If so, that's another term in your answer, otherwise, what's left is the remainder, if it isn't 0.

Answer 2

It's a long division problem. The book answer is x-y+z. But how do you get there doing the long division?

Answer 3

It would be pretty easy if it was just a poly / a monomial. But this thing was tricky to me.

Answer 4

Also I don't see how you could subtract the give denominator from the given numerator when there aren't any like terms on one versus the other.

Answer 5

*given

Answer 6

A little easier said that done huh?

Answer 7

You gotta have faith :-)

\[(x^2 - y^2 + 2yz - z^2) / (x + y - z)\]

Okay, by looking it, we know our answer has to have an x, a y, and a z, right? That's the only way we'll get x^2, y^2 and z^2 terms. We'll do this as a "speculative division" — we'll speculate about the answer, and then see if works. What we do will also bear some similarity to solving multiple equations with combination.

Let's make a little table of what {x, y, z} * (x+y-z) gives us:
\[x(x+y-z) = x^2 + xy - xz\]\[y(x+y-z) = y^2 + xy - yz\]\[z(x+y-z) = -z^2 + xz + yz\]
Those are our building blocks. How do we combine them to get our original numerator?
Well, we need an x in the quotient to produce the x^2 term. We need a -y in the quotient to produce the -y^2 term. We need a z in the quotient to produce the -z^2 term. Let's just add up those equations and see if we get the right thing!
\[x(x+y-z) - y(x+y-z) + z(x+y-z)=\]\[x^2+xy-xz - (y^2+xy-yz) + (-z^2+xz+yz)=\]
\[x^2 + xy - xz -y^2 -xy -(-yz) -z^2 +xz +yz=\]\[x^2-y^2 -z^2 + xy -xy +xz- xz +yz+yz \] and after the dust settles and we cross off all the terms that combine to 0, we have\[x^2-y^2+2yz -z^2\] which is our original numerator. Therefore, the quotient we "guessed", (x - y + z), is correct!

Answer 8

It took me a few minutes. Let me look at what you have there. Thanks for the help too.

Answer 9

I will have to test that theory on other problems, but thanks for that. I really appreciate it. However, do you know how to do it using the long division?

Answer 10

In other words, suppose I don't have a guess. Then what?

Answer 11

Well, it really *is* what you do when you do long division, I just organized the work a little differently.

Answer 12

Oh ok. I got you.

Answer 13

I will try that over the weekend on some other problems.

Answer 14

x - y + z
x+y-z x^2 - y^2 + 2yz - z^2
x^2 xy -xz
---------------------------------------------------
-y^2 + 2yz - z^2 - xy +xz
-y^2 +yz - xy
--------------------------------------------
yz - z^2 +xz
yz -z^2 +xz
---------------------------------

Answer 15

I tend to make mistakes when doing lots of algebra, so if I can think a little first and simplify the pencil-pushing, my accuracy goes up :-)

Answer 16

Writing it out like a long division problem, it took me a couple of tries before I avoided making any mistakes. Doing it the "combination" way, I did it the first time, and it took less time, too!

Answer 17

That was the first time I've seen it done that way. I am going to look that up on youtube later.

Answer 18

I agree it's pretty weird seeing all of these "other" terms popping up as you do the long division, but like I said, you just have to have faith that they will all go away in the end :-)

Answer 19

It kind of threw me for a loop, but I will have this figured out by tomorrow.

Answer 20

That's the attitude I like to hear! :-) Too many people just say "oh, this is hard" or "I'm dumb" or whatever. Much of this stuff isn't really hard at all — it just takes the right explanation, and practice.

Answer 21

Well, thanks again.

Answer 22

I'm a fan now. Ttyl.

Answer 23

Use PEMDAS!