Factoring Polynomials

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BlBl

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I've got a polynomial factoring problem that I'm finding tough. Usually I can just breeze through these without much thought (as they're typically just adding and multiplying) but this one really got me puzzled:

\(\displaystyle x^4-11x^2y^2+y^4\)

This is from a Schaum's pre-calculus book. I have the factors already, and multiplying them back gets the right answer, so I'm not worried about that. I just don't see what the first step in factoring this thing might be. How did they derive that answer they gave? GCF is out, grouping is out (or is it?), is this demonstrating completing the square, or am I missing something obvious?

I'm also going to copy this here in a non LaTex form:
x^4-11x^2y^2+y^4
 
Add and subtract \(\displaystyle 9x^{2}y^{2}\).

This results in a perfect square.

\(\displaystyle x^{4}-11x^{2}y^{2}+y^{4}+9x^{2}y^{2}-9x^{2}y^{2}\)

\(\displaystyle x^{4}-2x^{2}y^{2}+y^{4}-9x^{2}y^{2}\)

\(\displaystyle \underbrace{x^{4}-2x^{2}y^{2}+y^{4}}_{\text{perfect square trinomial}}-(3xy)^{2}\)

\(\displaystyle (x^{2}-y^{2})^{2}-(3xy)^{2}\)

Now, use the difference of two squares:

\(\displaystyle (x^{2}+3xy-y^{2})(x^{2}-3xy-y^{2})\)
 
Thank you for the reply Galactus ;-)

So it was a "complete the square" type problem after all. Now that I know that (not that I didn't really, as the book gave me the factors already) my question is how would I go about doing more of those? I seem to have difficulty recognizing the problem form above as a perfect square problem (multiple variables, no coefficient on the constant terms). And even when I guess to use the completing the square method I seem to have trouble picking the right number to do it with.

What came to mind first was to reduce the 11 term down to a 1 term, but looking at it now that wouldn't have made sense for 2ab. What was your first step in solving this?
 
If you had started by looking at the value of \(\displaystyle (x^2-y^2)^2\)

you would have had \(\displaystyle x^4-2x^2y^2+y^4\)

now note the difference between this and your starting equation ;)

The reason for my first equation?

Your equation had an x^4 a y^4 and a -ve in the middle

hth
 
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Thank you procyon. I mean, I can definitely see the relation now, but that's after the fact. Is there no way to get this before hand other than lots and lots of practice to become familiar with the forms of perfect squares in polynomial/variable arrangements?

When looking through completing the square tutorials online I come across suggestions like halving the middle term's coefficient and then squaring that quotient. That approach just never seemed to work with my problem above.
 
BlBl said:
lots and lots of practice
Click to expand...

That's the main ingredient ;)

As I said, your equation was \(\displaystyle x^4\ \ ????\ \ y^4 \)

which screams to me \(\displaystyle (x^2\ \ ????\ \ y^2)^2\)

if that went nowhere I'd have looked at \(\displaystyle (x\ \ ????\ \ y)^4\)

The next part was the sign for ????

Just keep it the same as what is already there

so you get \(\displaystyle (x^2-y^2)^2\)

hth
 
:D

Thanks again. I guess I've just gotta try and find a place that has lots and lots of completing the square problems like the one above. I'll look around, but what I've found so far has tended to focus on \(\displaystyle (\frac{b}{2})^2\), and using even numbered coefficients in the middle monomial, rather than a guess search for the right number.

But practice, practice, practice, I will.
 
BlBl said:
Thank you for the reply Galactus ;-)

So it was a "complete the square" type problem after all. Now that I know that (not that I didn't really, as the book gave me the factors already) my question is how would I go about doing more of those? I seem to have difficulty recognizing the problem form above as a perfect square problem (multiple variables, no coefficient on the constant terms). And even when I guess to use the completing the square method I seem to have trouble picking the right number to do it with.

What came to mind first was to reduce the 11 term down to a 1 term, but looking at it now that wouldn't have made sense for 2ab. What was your first step in solving this?
Click to expand...


It's not so much a completing the square as perfect square trinomial.

The idea is to add and subtract a term that results in a perfect square, then use the difference of two squares.

Take \(\displaystyle 36x^{4}+15x^{2}+4\)

Add and subtract \(\displaystyle 9x^{2}\)

\(\displaystyle (36x^{4}+24x^{2}+4)-(9x^{2})\)

\(\displaystyle (6x^{2}+2)^{2}-(3x)^{2}\)

Difference of two squares:

\(\displaystyle (6x^{2}+3x+2)(6x^{2}-3x+2)\)
 
procyon said:
...note the difference between this and your starting \(\displaystyle > > \)equation\(\displaystyle < < \) ;)

The reason for my first \(\displaystyle > > \)equation?\(\displaystyle < < \)

Your \(\displaystyle > > \)equation \(\displaystyle < < \)had an x^4 a y^4 and a -ve in the middle

hth
Click to expand...
procyon said:
As I said, your \(\displaystyle > > \)equation\(\displaystyle < < \) was \(\displaystyle x^4\ \ ????\ \ y^4 \)
Click to expand...


procyon,

we don't have equations here; we have expressions. More specifically, they
are polynomial expressions.

.
 
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Speaking of expressions… is there a name for this type of thing? As Galactus notes, this isn't really "completing the square" so looking up examples of completing the square won't give you things that help.

I've got more (I think 5 more) problems that are giving me trouble in this chapter. I'll put them up later.
 
Alright here's a factoring problem:

\(\displaystyle x^2-6x+9-y^2-2yz-z^2\)

Now I wrote down for this \(\displaystyle (x-3)^2-(y-z)^2\)

But the book has something else. Thoughts?
 
BlBl said:
Alright here's a factoring problem:

\(\displaystyle x^2-6x+9-y^2-2yz-z^2\)

Now I wrote down for this \(\displaystyle (x-3)^2-(y-z)^2\)

But the book has something else. Thoughts?
Click to expand...

Rewrite it as


\(\displaystyle x^2 - 6x + 9 - (y^2 + 2yz + z^2)\)


Please continue. . .
 
BlBl said:
Alright here's a factoring problem:

\(\displaystyle x^2-6x+9-y^2-2yz-z^2\)

Now I wrote down for this \(\displaystyle (x-3)^2-(y-z)^2\)

But the book has something else. Thoughts?
Click to expand...

My thoughts.....

the pattern for factoring a difference of two squares is this:

\(\displaystyle a^2 - b^2 = (a + b)(a - b)\)

You've got: \(\displaystyle (x-3)^2-(y-z)^2\)

write it as \(\displaystyle [(x - 3) + (y - z)]*[(x - 3) - (y - z)]\)

If you want to eliminate the "inner" parentheses, you can:

\(\displaystyle [x - 3 + y - z]*[x - 3 - y + z]\)
 
I realized later that I'd just disregarded the difference of two squares pattern. These elementary mistakes are costing me :-(

Alright, I'll put up a few more that have been giving me trouble. I feel like I do better just by writing them out and communicating them to others. Thanks folks!
 
Ok, here's a few more (all factoring):

\(\displaystyle P+Pr+(P+Pr)r+[P+Pr+(P+Pr)r]r\)

\(\displaystyle 3(5x+2)^2(5)(3x-4)^4+(5x+2)^3(4)(3x-4)^3(3)\)

\(\displaystyle 5(x^2+4)^4(8x-1)^2(2x)+2(x^2-4)^5(8x-1)(8)\)

Alright for the first I assumed that the GCF here was \(\displaystyle P+Pr+(P+Pr)r\). With that as the case I proceeded to write \(\displaystyle P+Pr+(P+Pr)r•(1+r)\). But that wasn't what the book wrote as an answer. In looking back at it, I suppose I should probably add up all the like terms first and see what happens.

For the second I assumed that the GCF was \(\displaystyle (5x+2)^2(4)(3x-4)^3\). With that I… was stuck. I didn't know what to do with those weird numbers off by themselves in parentheses for this one and the next one.

Help?

These might be more tedious than difficult.
 
BlBl said:
Ok, here's a few more (all factoring):

\(\displaystyle P+Pr+(P+Pr)r+[P+Pr+(P+Pr)r]r\)

= P(1+r) + Pr(1+r) +Pr(1+r) + P(1+r)r2


= P(1+r)[1 + 2r + r2 ]

= P(1+r)3

\(\displaystyle 3(5x+2)^2(5)(3x-4)^4+(5x+2)^3(4)(3x-4)^3(3)\)

\(\displaystyle 5(x^2+4)^4(8x-1)^2(2x)+2(x^2-4)^5(8x-1)(8)\)

Alright for the first I assumed that the GCF here was \(\displaystyle P+Pr+(P+Pr)r\). With that as the case I proceeded to write \(\displaystyle P+Pr+(P+Pr)r•(1+r)\). But that wasn't what the book wrote as an answer. In looking back at it, I suppose I should probably add up all the like terms first and see what happens.

For the second I assumed that the GCF was \(\displaystyle (5x+2)^2(4)(3x-4)^3\). With that I… was stuck. I didn't know what to do with those weird numbers off by themselves in parentheses for this one and the next one.

Help?

These might be more tedious than difficult.
Click to expand...
.
 
Thanks Subhotosh Khan. Now I see how I should've gone forward with the problem. Lookagain, I'll start a new thread. I didn't want to be perceived as a spammer, and seeing as two of my examples had already been addressed, I figured I'd just continue.

Ah well.
 
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