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In brief, rational expressions are algebraic expressions that can be expressed as quotient of polynomials. For example, `x^2` and `(x^3 - 4)/(2 + x^2)` are rational expressions.` `

Manipulation of these expressions are simple -- you manipulate them as you manipulate fractions!

Now, let's look at multiple rational expressions, and try to do the aforementioned operations on them. In the following examples, we'll leave out polynomial equations (which are technically also rational expressions) as they are a special case (easy to manipulate).

1. Addition and Subtraction

Like adding fractions, you first have to find the common denominator before adding them -- that is, unless they have the same denominator, you can't add them right away. Take for example:

`(2/(x+5)) + (x/(x^2 + 1))`

Both are rational expressions. To add them, we first have to look for the common denominator. The easiest way would simply be to multiply the denominator (unless of course they have a common factor) -- in this case: `(x+5)(x^2 + 1)` .

Hence:

`(2/(x+5)) + (x/(x^2 + 1)) = (2*(x^2 + 1))/((x+5)*(x^2 + 1)) + (x*(x+5))/((x+5)*(x^2 + 1))`

The sum then, can be easily calculated, then simplified. The same is done with subtraction (since you can look at subtraction as adding the negative... so no additional step for this operation).

2. Multiplication

Again, multiplication of rational numbers is also like the process done for fractions. Multiply the numerator of one by that of the other, then do the same for the denominator:

`(5/(x+2))*(x/(x+1)) = (5x)/((x+2)(x+1))`

Then, simply simplify, and cancel terms that can be cancelled (in this case, we can't cancel anything). Here's an example where cancellation can be done:

`((15x^2y)/(x^5))*((5y)/(3yx^4))= ((15*5*x^2*y*y)/(3*x^5*y*x^4))`

Notice that we can simplify it further by cancelling/dividing stuff -- e.g. `15/3 = 5.`

In the end, this can be simplified to:

`((25y)/(x^7))`

3. Division

Rational expressions (those we've used here) are actually already quotients of other rational expressions (particularly of polynomial equations). For instance, `(x/(x^2+1))` is a quotient of the expression `x` , and the expression `x^2 + 1` . Let's have another example:

`(x^2 + 2x + 1)/(x+5) div (x+1)/(x^2 - 25)`

This (like any division) is just like multiplying the first expression by the reciprocal of the second (and so we reduce the problem into multiplication):

`((x^2 + 2x + 1)/(x+5)) * ((x^2-25)/(x + 1))`

However, we're going to use other techniques involved (just to demonstrate them) in simplifying this. One of the most useful techniques when dealing with rational expressions (or polynomial expressions) and the simplification of these algebraic expressions is factoring.

Notice that the numerator of the first term can be factored as: `(x+1)^2.`

Meanwhile, the numerator of the second term can be factored as: `(x+5)(x-5)` .

Now, simplification is easier since we see that a lot of terms cancel -- in particular, `(x+1)` and `(x+5)` .` `

Hence, the simplified quotient is:

`(x+1)(x-5) = x^2 - 4x - 5`

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Again, manipulation of rational expressions is not different from manipulating numerical fractions. The only difference, probably, is that you can 'factor' (in a somewhat different sense; since we are also doing this for numbers -- when dividing 15 by 3 for instance, we are 'implicitly' factoring 15 as a product of 5 and 3, and consequently cancelling 3) to simplify your expression.

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