Math Help - Question on Factoring

1. Question on Factoring

I am uncertain how to factor this expression correctly.

$12x^2 y^3 z + 18x^3 y^2 z^2 - 24xy^4 z^3$

The greatest common factor is $6xy^2 z$

However, I am not seeing how the author picks the 2nd power when factoring out from the rest of the expression. I was under the impression that when factoring out powers of a variable, the smallest power that appears in any on term is the most that can be factored out.

In this example I am not seeing how $6xy^2 z$ is the smallest power that appears in the terms if someone could please clarify. Specifically why "xy" is raised to the 2nd power for the lowest.

2. Originally Posted by cmf0106
I am uncertain how to factor this expression correctly.

$12x^2 y^3 z + 18x^3 y^2 z^2 - 24xy^4 z^3$

The greatest common factor is $6xy^2 z$

However, I am not seeing how the author picks the 2nd power when factoring out from the rest of the expression. I was under the impression that when factoring out powers of a variable, the smallest power that appears in any on term is the most that can be factored out.

In this example I am not seeing how $6xy^2 z$ is the smallest power that appears in the terms if someone could please clarify. Specifically why "xy" is raised to the 2nd power for the lowest.
$12x^2 y^3 z + 18x^3 y^2 z^2 - 24xy^4 z^3$

You want to take out as much as you can from that equation, but you can't take out more than the equation consists of.

$6xy^2 z$

6 is the most we can take out of the original equation, and still have integers left for coefficients.

For the $x$
Note that the smallest $x$ we have in the original equation is to the degree 1, se we need to take out an x of degree 1.

For the $y^2$ (your source of confusion)
Remember what i said in the beginning, take out as much as you can, but not more than there is in the equation.

So we have $y^4 ; y^2 ; y^3$
The most we can take out there is a $y^2$. If we, for instance, take out $y^3$, our $y^2$ will become $y^{-1}$ which only leaves us with a bigger mess.

If we only take out $y$, we are also just keeping things complicated. Why only take the $y$, when you can remove more from the equation to make it even simpler?

For the $z$

We cannot remove anything more than a $z$ to degree 1 from the equation. (Technically we can, but that doesn't mean we should )

3. Ah thank you very much! I did not realize I could simply look through x,y, & z and see what the greatest power for each that could be removed was.

4. Originally Posted by cmf0106
Ah thank you very much! I did not realize I could simply look through x,y, & z and see what the greatest power for each that could be removed was.
You are welcome

5. Similarly, in the same book the author is trying to show that if you miss the largest GCF the first time through you are still in good standing.

However for this particular expression: $484x^3 y^2 + 132x^2 y^3 - 88x^4 y^5$

the author demonstrates assuming you determined that the GCF of the expression in this example is $4x^2 y$

However, shouldn't the correct assumed GCF be $4x^2 y^2$? Listing the Y powers in increasing order for this expression: $y^2, y^3, y^5$. Since the lowest power of y is y^2 wouldnt the correct assumption be $4x^2 y^2$?

6. Originally Posted by cmf0106
Similarly, in the same book the author is trying to show that if you miss the largest GCF the first time through you are still in good standing.

However for this particular expression: $484x^3 y^2 + 132x^2 y^3 - 88x^4 y^5$

the author demonstrates assuming you determined that the GCF of the expression in this example is $4x^2 y$

However, shouldn't the correct assumed GCF be $4x^2 y^2$? Listing the Y powers in increasing order for this expression: $y^2, y^3, y^5$. Since the lowest power of y is y^2 wouldnt the correct assumption be $4x^2 y^2$?
Well yes you are in good standing. You should be able to factor out the rest, by inspection soon afterwards.

And yes the correct GCF should be $4x^2 y^2$ !

7. Originally Posted by janvdl
Well yes you are in good standing. You should be able to factor out the rest, by inspection soon afterwards.

And yes the correct GCF should be $4x^2 y^2$ !
So by my conclusion and your affirmation, should be the correct GCF not $4x^2 y$. So I have found an error in Algebra for dummies ?

8. Originally Posted by cmf0106
So by my conclusion and your affirmation, should be the correct GCF not $4x^2 y$. So I have found an error in Algebra for dummies ?
It could be. The greatest CGF is that, but he might be trying to explain a certain concept, or maybe mistakes one can make by assuming the wrong GCF.

9. Alright thanks again. Also the whole reason I am going back and reading up on algebra is because I plan on teaching myself calculus; however, calculus is rooted in algebra or so I am told.

With that being said, the later chapters of "Algebra I for Dummies" covers much of what I did previously freshman year of College Algebra. Do you think I should read up on algebra II material before I teach myself pre-calculus? Or perhaps Algebra II books are not required if the one I am currently reading meshes up with my old college algebra class quite well.

Thanks!

10. Originally Posted by cmf0106
Alright thanks again. Also the whole reason I am going back and reading up on algebra is because I plan on teaching myself calculus; however, calculus is rooted in algebra or so I am told.

With that being said, the later chapters of "Algebra I for Dummies" covers much of what I did previously freshman year of College Algebra. Do you think I should read up on algebra II material before I teach myself pre-calculus? Or perhaps Algebra II books are not required if the one I am currently reading meshes up with my old college algebra class quite well.

Thanks!
Calculus I also makes use of some basic trigonometry.

Doing Algebra II too is a very good idea.