How to reduce rational expressions

X^5 - 2x^4 - 15x^3

Question

How to reduce rational expressions

X^5 - 2x^4 - 15x^3

debbieg · Answer

Well, that isn't a rational expression. But if that were, say, in the numerator of a rational expression, you would want to factor it. First factor out the GCF (which is a power of x). Then you'll have a trinomial which factors easily to the product of two binomials ("undo" the FOIL).

atlas · Answer

The rational expression has 3 terms x^5, -2x^4 and -15x^3. All of them have x^3 common
$$x ^{3}(x ^{2}-2x -15)$$
The right hand side can be simplified to write: (This is done by factorizing the constant term (15=5*3))
$$x ^{3}(x ^{2}-5x+3x -15)$$
$$x^{3}(x(x-5)+3(x-5))$$
(x-5) is again common to two expressions...So:
$$x^3 (x-5)(x+3)$$
This is the final simplified form

debbieg · Answer

Yes, like I said above, this POLYNOMIAL can be FACTORED (I was waiting to see if the student could do that himself, but oh well).

But this is NOT a "rational expression". It's a polynomial. Factoring a polynomial doesn't "simplify" it, it just factors it.

A rational expression is a ratio of 2 polynomials.

atlas · Answer

Well I guess I should have let the student solve it but I thought he might be solving this for the first time and so he could see how it is done.
And I believe polynomials are special form of rational expressions with denominator 1 :)

debbieg · Answer

Hah, granted that would be strictly correct. :) 

But when the student's question is "how to REDUCE a rational expression", I think there is something missing from the problem as stated if we are only given a polynomial. The student should not be confused into thinking that "factoring" a polynomial, "reduces" it. :)

atlas · Answer

Guess you are right.........