Factoring polynomials with fractional and negative exponents.

Question

I’m having trouble understanding factoring polynomials with fractional and negative exponents. For example:

x(x+1)^(-3/4) + (x+1)^(1/4)

I recognize you have to factor it but the part I get confused on is why do you take the (-3/4) exponent when factoring. I understand it’s smaller than 1/4 but is it a factor?!? If you can help clarify this I would really appreciate it. 🙂

In: 2

Answer 1

Negative exponents aren’t fun (fractional or not, polynomial or not) because they mean we secretly have an uncommon/unshared denominator. Really, this isn’t any different than simplifying mixed fractions:

> 3(5)^-2 + (5)^1 = 3/25 + 5/1

How do we deal with an unshared denominator in this basic case? We make sure all the other terms in a sequence have matching denominators by “multiplying-by-1” (where in this case “1=25/25”):

> 3/25 + (5/1)(25/25)= 3/25 + 125/25 = (3)(1/25)+(125)(1/25) = (1/25)(3+125)

So what do we do with your polynomial expression?

> x(x+1)^-a + (x+1)^b = x/(x+1)^a + (x+1)^b

We start off by “multiplying-by-1” (where in this case “1=(x+1)^a / (x+1)^a”) the other terms in the sequence:

> x/(x+1)^a + (x+1)^b = x/(x+1)^a + ((x+1)^b )((x+1)^a / (x+1)^a ) = x/(x+1)^a + (x+1)^b+a /(x+1)^a

And then we just divide out the common denominator:

> x/(x+1)^a + (x+1)^b+a /(x+1)^a = (1/(x+1)^a )(x + (x+1)^b+a )

And then hopefully you can see how plugging back in your a=3/4 and b=1/4 gets to the answer.

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The reason why we “factor out a -3/4” (rather than “factor out a 1/4) is simply because it is the one term that has a troublesome secret denominator.