Reduce the following rational expression to lowest terms.
x^{2} - 2 x - 8/ x^{2} - 9 x + 20
Solution
When reducing a rational expression to lowest terms the first thing that we will do is factor both the numerator and denominator as much as possible. That ought to always be the first step in these problems.
Also, the factoring in this section, & all successive section for that matter, will be done with no explanation. It will be supposed that you are capable of doing and/or verifying the factoring on your own. In other terms, ensure that you can factor!
x^{2 }- 2x - 8 /x^{2} - 9 x + 20
First we'll factor things out as fully as possible. Remember that we can't cancel out anything at this instance in time as every term contain a "+" or a "-" on one side of it! We've got to factor
x^{2} - 2 x - 8 /x^{2} - 9 x + 20 = ( x - 4) ( x + 2)/( x - 5) (x - 4)
At this instance we can see that we've got a common factor in the numerator and the denominator both and so we can cancel out the x-4 from both. Doing this gives,
x^{2} - 2 x - 8 /x^{2} - 9 x + 20 = x + 2 /x - 5
It is also all the farther that we can go. Nothing else will cancel out and thus we have decreased this expression to lowest terms.
In other terms, a minus sign in front of a rational expression can be moved over the whole numerator or whole denominator if this is convenient to do that. However, we ought to be careful with this. Let the following rational expression.
- x + 3 /x + 1
In this case the "-" onto the x can't be moved to the front of the rational expression as it is only on the x. To move a minus sign to the front of a rational expression it has to be times the whole numerator or denominator. Thus, if we factor a minus out of the numerator then we could move it into the front of the rational expression as follows,
- x + 3 /x + 1 = - ( x - 3) / x + 1= -(x-3)/(x+1)
Here, the moral is that we have to be careful with moving minus signs around in rational expressions.
Now we need to move into adding, subtracting, multiplying & dividing rational expressions. Let's begin with multiplying & dividing rational expressions. The general formulas are such as,
(a/b) ⋅ (c /d)= ac /b d
(a/b) /(c /d)=(a/b)÷(c/d)=(a/b).(d/c)
Note the two distinct forms for mentioning division. We will employ either as required so ensure you are familiar with both. Notice as well that to do division of rational expressions all that we have to do is multiply the numerator by the reciprocal of the denominator (that means the fraction along with the numerator & denominator switched).
There are a couple of special cases of division that we have looked at. Generally above both the numerator and the denominator of the rational expression where fractions, though, what if one of them isn't fraction. Thus let's look at the following cases.
a/(c/d) (a/b)/c
Initially Students frequently make mistakes with these. To properly deal with these we will turn the numerator (first case) or denominator (second case) into fraction and then do the general division on them.
a/(c/d)=(a/1)/(c/d)=(a/1).(d/c)=(ad/c)
(a/b)/c =(a/b)/(c/1)=(a/b).(1/c)=(a/bc)
Be careful with these cases. It is simple to make a mistake with this case and do the division incorrectly.