My textbook simplifies this expression but..?

Question

I don't see where the x^2 in the denom has gone.
x+1/14x^4-5/(x^3-3x^2)

They say the LCD is 14x^4(x-3)
but where has the x^2 'gone'?

Answer 1

x^2 is a factor in 14x^4, so inserting x^2 as an additional factor in LCD is not needed and not allowed. One requirement for least common denominator LCD is that it be divisible by all the denominator factors with no remainder, but a second requirement, that applies in your question, is that LCD have no factor that is redundant, that is not needed.

Details in finding LCD for this problem: First you have to factor the last denominator to get all the simplest factors, one being x^2. Since 14x^4 must be a factor in LCD, and since 14x^4 is divisible by x^2 with no remainder, then LCD does not need factor x^2 and indeed must not have factor x^2, because then it would no longer be a LEAST common denominator (since x^2 would go into it twice). (Restating previous points, but in slightly different way.)

Here is the factorization into simple denominators:

x + [1/(14x^4)] - [5/(x^3-3x^2)]

Expand 1st term and factor last term:

[x/1] + [1/(14x^4)] - [5/(x-3)(x^2)]

Denominators cannot be further factored. Now select those factors from denominators that must be present in order for all factors in the denominators of all terms to be divisors without remainders.

Method 1: The denominator factors are 1, 14x^4, (x-3), and x^2. Go down the list: Do not select 1 since 1 is always implied, and 1 divides into any LCD without remainder, Select 14x^4 the first non-trivial denominator (really 2 factors, 14 and x^4). Select (x-3), since it does not divide into pending the LCD (i.e., what you have selected so far, 14x^4). Do NOTselect x^2 since it divides into the pending LCD without remainder, namely into x^4, to give quotient x^2, remainder 0. That completes finding LCD: (14x^4)(x-3).

Method 2: Suppose you had started from the right selecting factors for LCD, which is allowed (any order is allowed). Then the list of factors encountered would be x^2, (x-3), 14x^4, and 1. If selecting blindly without looking ahead the process would be this: Select x^2, then (x-3), then 14x^4. But at this point check previous selections to see if they are now redundant (not needed). (x-3) is still needed, but x^2 first chosen is now redundant because now x^4 is present. As before, not only is x^2 not needed for LCD, it is not allowed. That completes finding LCD: (x-3)(14x^4) (different order, equivalent answer).

During the selection process, after each selection of another factor for LCD, review all previous factors to see if any of them are now redundant, and if so, remove them.