Algebra help please

Question

Find the least common denominator (LCD) of the following rational fractions:

Answer 1

LCD: (x + 1)² ( x + 5)( x - 3)

Answer 2

We take each of the denominators and break them down into their individual factors.

The first gives (x+5), (x-3), (x+1)

The second gives (x-3), (x+1)

The third gives (x+1)² or (x+1)(x+1)

In order to find the least common denominator, we take the product of all of these pieces, but if there are repeats across the different fractions, we do not repeat. So, (x+5)(x-3)(x+1) is the first. The second has (x-3), but we already have that piece in the first part. Then we have (x+1) which is also already in the first part. Finally, we come to the third fraction, and notice that we have an (x+1), but we don't have the second (x+1) included. So our final answer is (x+5)(x-3)(x+1)(x+1), which simplified gives (x+5)(x-3)(x+1)².

Answer 3

Mel, it's just like regular fractions. You break each denominator down to its prime factors and use one of each.

1/4, 1/12, 1/15

4 = 2²

12 = 2²(3)

15 = 3(5)

We are going to use each prime factor the most times it shows up in one denominator.

2² is in two denominators, so we use it.

3 shows up in two denominators, so we use it.

5 shows up in one denominator, so we use it.

LCD = 2²(3)(5)

Later, when we convert our fractions, we multiply the numerator and denominator by what's missing from the numerator.

1(3)(5))/4(3)(5) + 1(5)/12(5) + 1(2²)/15(2²) = 15/60 + 5/60 + 4/60 = 24/60 = 2³(3)/2²(3)(5) = 2/5

Now, in your problem, we have x + 5 show up once, so we use it.

x - 3 is in two denominators, so we use it.

x + 1 shows up in all three, but it's squared in the 3rd, so we use (x + 1)².

Sooooo, our LCD is (x + 5)(x - 3)(x + 1)².

2(x + 1)/(x + 5)(x - 3)(x + 1)² + 4x(x + 5)(x + 1)/(x + 5)(x - 3)(x + 1)² + (x - 1)(x + 5)(x - 3)(x + 1)/(x + 5)(x - 3)(x + 1)²

I know that's more than you asked for, but I hope it helps!