Example #2
(x - 2)/(x + 5) + (x^2 + 5x + 6)/(x^2 + 8x + 15)
First, we need to factor.
x^2 + 5x + 6 = (x + 3)(x + 2)
x^2 + 8x + 15 = (x + 5)(x + 3)
After we replace the factored terms, our new expressions looks like:
(x - 2)/(x + 5) + (x + 3)(x + 2)/(x + 5)(x + 3)
To find our common denominator, we simply write down our denominators. From the first term, we have (x+ 5) as our denominator. In the second term, we have (x + 5) and (x + 3). Since we already have (x + 5) written as part of our common denominator, we will just write (x + 3). So, our common denominator is (x + 5)(x + 3).
Our next step is to multiply each piece of the expression, so we have (x + 5)(x + 3) as our new denominator. In the first fraction, we need to multiply by (x + 3) over (x + 3). This will give us (x - 2)(x + 3)/(x + 5)(x + 3) as our first fraction. Looking at the second fraction, I notice I already have (x + 5)(x + 3) in the denominator, so I can leave this one alone.
Now, let's write the entire numerator over our common denominator.
((x - 2)(x + 3) + (x + 3)(x + 2))/(x + 5)(x + 3)
Let's simplify the numerator by writing the numerator over our common denominator and FOIL.
(x - 2)(x + 3) = (x^2 + x - 6) and
(x + 3)(x + 2) = (x^2 + 5x + 6)
Collect like terms in the numerator.
2x^2 + 6x
Factor the numerator if possible.
2x(x + 3)
Our expression now looks like:
2x(x + 3)/(x + 5)(x + 3)
We can slash, or cancel, (x + 3) over (x + 3).
This gives us our final answer, 2x/(x + 5).
Example #3
(x^2 + 12x + 36)/(x^2 - x - 6) + (x + 1)/(3 - x)
First, we need to factor.
(x^2 + 12x + 36) = (x + 6)(x + 6)
(x^2 - x + 6) = (x - 3)(x + 2)
After we replace the factored terms, our new expressions looks like:
(x + 6)(x + 6)/(x - 3)(x + 2)) + (x + 1)/(3 - x)
To find our common denominator, we simply write down our denominators. From the first term, we have (x - 3)(x + 2) as our denominator. In the second term, we have (3 - x). I could write (3 - x) as part of the common denominator, but I know that -1 * (x - 3) = (3 - x). So, now it will match with the denominator (x - 3).
Now, our expression looks like:
(x + 6)(x + 6)/(x - 3)(x + 2)) + (x + 1)/-1(x - 3
And that -1? It can be put into the numerator. Remember, 1/-1 = -1/1 = -1. It doesn't matter where I put the -1 in the fraction as long as I have a +1 to match it.
So, our common denominator is (x - 3)(x + 2).
In the first fraction, I already have the common denominator (x - 3)(x + 2), so I leave that one alone. In the second fraction, I need to multiply by (x + 2) over (x + 2). This gives us the common denominator of (x - 3)(x + 2).
Our expression now looks like:
(x + 6)(x + 6)/(x - 3)(x + 2) + (-1)(x + 1)(x + 2)/(x - 3)(x + 2)
Let's simplify the numerator by writing the numerator over our common denominator and using FOIL, which is First Outside Inside Last.
(x + 6)(x + 6) = x^2 + 12x + 36
and
(-1)(x + 1)(x + 2) = (-1)(x^2 + 3x + 2) = -x^2 - 3x- 2
Collect like terms in the numerator. Our expression now looks like:
(9x + 34)/(x - 3)(x + 2)
The numerator doesn't factor, so our last step is to FOIL the denominator.
Our final answer is (9x + 34)/(x^2 - x - 6).