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I'm currently reviewing college algebra and I'm learning about factoring polynomials and algebraic expressions.

I have no difficulties finding the GCF of algebraic expressions whose variables have positive integer exponents, but I don't understand the process when it comes to algebraic expressions whose variables have negative exponents.

I understand why you factor out the power of each of the variables with the smallest exponent when working with positive exponents, but I don't see why that rule applies when dealing with fractional and negative exponents. Actually, I don't have difficulty seeing why the rule applies to positive fractional exponents because, for instance, I can see that $$3x^{3/2}-9x^{1/2}+6x^{1/2}=3(x^{1/2})^3-9(x^{1/2})^1+6(x^{1/2})^1$$ and so I can see that the GCF is $3(x^{1/2})^1=3x^{1/2}$. But admittedly, I did not see this expression and know intuitively that the GCF should be the expression that contained the power of $x$ with the smallest exponent. It wasn't until I rewrote the expression as above that I saw that why the rule makes sense.

How can I rewrite expressions involving variables with negative exponents to see why the rule is still valid? If the terms in the above example were instead $$3x^{2/7}-9x^{-3/4}+6x^{-3/5}$$ would the GCF be $3x^{-3/4}$ as the rule would suggest?

What is the definition of greatest common factor I should keep in mind when dealing with algebraic expressions and polynomials?

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  • $\begingroup$ Not sure how common it is to call it GCF, but in this context I assume it would always be the smallest exponent. The point of the exercise is that after factoring out the GCF you would be left with an expression that contains a constant term without $x$, and all other terms would contain positive powers of $x$. $\endgroup$
    – dxiv
    May 8 at 2:23
  • $\begingroup$ Yes, I agree, definitely that the point of the exercise would be to end up with an expression that contains a constant term and all other terms having positive powers of $x$. And I also understand the mechanics of doing so, but what I do not see is why this rule extends to negative exponents. What definition of greatest common factor should I keep in mind when working with polynomials and other algebraic expressions? $\endgroup$
    – hdngnmyrm
    May 8 at 2:35
  • $\begingroup$ The smallest exponent is the only choice that satisfies both conditions. In your example $3x^{2/7}-9x^{-3/4}+6x^{-3/5}$ $= x^{-3/4}\left(3x^{2/7+3/4}-9x^{-3/4+3/4}+6x^{-3/5+3/4}\right)$ $= x^{-3/4}\left(3x^{29/28}-9+6x^{3/20}\right)$. If you chose any factor other than $x^{-3/4}$ then you would not have a constant term, or would have negative powers of $x$, or both. $\endgroup$
    – dxiv
    May 8 at 2:45
  • $\begingroup$ Makes sense now. Thanks $\endgroup$
    – hdngnmyrm
    May 9 at 1:25
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It hinges on the question of whether you consider $x^{-3/5}$ to be a factor of $x^{2/7}$, which I suspect varies from source to source. The advantage of being able to factor out a term with negative exponent like this is that it leaves a polynomial with positive exponents, which you're more likely to be able to do something with.

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I am writing an answer to my own question because it might help someone else see why finding the GCF of algebraic expressions involves taking out powers of each of the variables with the smallest exponent, whether they are positive, negative, or fractional and to feel justified in doing so. As an example, an algebraic expression such as $(2+x)^{-2/3}x + (2+x)^{1/3}$ can be rewritten as follows:

$(2+x)^{-2/3}x + (2+x)^{1/3} = (2+x)^{-2/3}x + (2+x)^{-2/3}(2+x)$

You should now be able to clearly see that we can factor $(2+x)^{-2/3}$. If not, consider

$ux + u(2+x)$, where $u = (2+x)^{-2/3}$. After distributing, we have

$ux + 2u +ux = 2ux + 2u$.

The GCF is $2u$ and the factored expression is $2u(x + 1)$ and after substituting for $u$, it becomes $2(2+x)^{-2/3}(x + 1)$.

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When looking at the example \begin{align*} 3x^{3/2}-9x^{1/2}+6x^{1/2}=\color{blue}{3x^{1/2}}\left(x-3+2\right) \end{align*} we factor out $x^{p/q}$ such that (at least) one term has no more factor $x$ and all other terms have factors $x^t$ with non-negative exponents $t\geq 0$.

We can do the same with the other example \begin{align*} 3x^{2/7}-9x^{-3/4}+6x^{-3/5}&=3x^{40/140}-9x^{-105/140}+6x^{-84/140}\tag{1}\\ &=\color{blue}{3x^{-105/140}}\left(x^{145/140}-3+2x^{21/140}\right)\tag{2}\\ &=\color{blue}{3x^{-3/4}}\left(x^{29/28}-3+2x^{3/20}\right) \end{align*}

Comment:

  • In (1) we conveniently use a representation with the least common multiple of the denominators of $\frac{2}{7}, -\frac{3}{4}$ and $-\frac{3}{5}$, i.e. with the $\mathrm{lcm}(7,4,5)=140$.

  • In (2) we can now factor out the GCF $3x^{-105/140}$ and simplify the exponents in the last step.

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