Algebraic fractions (or rational functions) can seem extremely complex at first glance and absolutely impossible to solve in the eyes of a student who does not know them. It is difficult to understand where to start by looking at the set of variables, numbers and exponents; luckily, however, the same rules apply that are used to solve normal fractions, such as 15/25.
Steps
Method 1 of 3: Simplify the Fractions
Step 1. Learn the terminology of algebraic fractions
The words described below will be used throughout the rest of this article and are very common in problems involving rational functions.
- Numerator: the upper portion of the fraction (for example (x + 5)/ (2x + 3)).
- Denominator: the lower portion of the fraction (e.g. (x + 5) /(2x + 3)).
- Common denominator: is the number that perfectly divides both the numerator and the denominator; for example, considering the fraction 3/9, the common denominator is 3, since it divides both numbers perfectly.
- Factor: a number that, when multiplied by another, makes it possible to obtain a third; for example, the factors of 15 are 1, 3, 5, and 15; the factors of 4 are 1, 2 and 4.
- Simplified equation: the simplest form of a fraction, equation or problem that is obtained by eliminating all common factors and grouping the similar variables together (5x + x = 6x). If you cannot proceed with further mathematical operations, it means that the fraction is simplified.
Step 2. Review the method for solving simple fractions
These are the exact steps you need to use to simplify the algebraic ones as well. Consider the example of 15/35; to simplify this fraction, you need to find the common denominator which, in this case, is 5. By doing so, you can eliminate this factor:
15 → 5 * 3
35 → 5 * 7
Now you can to delete similar terms; in the specific case of this fraction, you can cancel the two "5" and leave the simplified fraction 3/7.
Step 3. Remove the factors from the rational function as if they were normal numbers
In the previous example, you could easily eliminate the number 5, and you can apply the same principle in more complex expressions, such as 15x - 5. Find a factor that the two numbers have in common; in this case it is 5, since you can divide both 15x and -5 by this very figure. As in the previous example, remove the common factor and multiply it by the "remaining" terms:
15x - 5 = 5 * (3x - 1) To verify the operations, multiply 5 again by the rest of the expression; you will get the numbers you started from.
Step 4. Know that you can eliminate complex terms just like simple ones
In this kind of problem, the same principle applies as for common fractions. This is the most basic method of simplifying fractions when calculating. Consider the example: (x + 2) (x-3) (x + 2) (x + 10) Notice that the term (x + 2) is present both in the numerator and in the denominator; accordingly, you can delete it just like you deleted the 5 from 15/35: (x + 2) (x-3) → (x-3) (x + 2) (x + 10) → (x + 10) These operations lead you to the result (x-3) / (x + 10).
Method 2 of 3: Simplify Algebraic Fractions
Step 1. Find the factor common to the numerator, the top of the fraction
The first thing to do when "manipulating" a rational function is to simplify each part that composes it; start with the numerator, dividing it into as many factors as possible. Consider this example: 9x-315x + 6 Start with the numerator: 9x - 3; you can see that there is a common factor for both numbers and it is 3. Proceed as you would any other number, "taking out" the 3 from the brackets and writing 3 * (3x-1); by doing so, you get the new numerator: 3 (3x-1) 15x + 6
Step 2. Find the common factor in the denominator
Continuing with the previous example, isolate the denominator, 15x + 6 and look for a number that can perfectly divide both values; in that case, it is the number 3, which allows you to rephrase the term as 3 * (5x +2). Write the new numerator: 3 (3x-1) 3 (5x + 2)
Step 3. Delete similar terms
This is the stage where you proceed to the true simplification of the fraction. Delete any number that appears in both the denominator and the numerator; in the case of the example, delete the number 3: 3 (3x-1) → (3x-1) 3 (5x + 2) → (5x + 2)
Step 4. You need to understand when the fraction is reduced to its lowest terms
You can affirm this when there are no other common factors to be eliminated. Remember that you cannot delete those that are in the brackets; in the previous problem, you can't delete the "x" variable of 3x and 5x, since the terms are actually (3x -1) and (5x + 2). As a result, the fraction is completely simplified and you can annotate the result:
3 (3x-1)
3 (5x + 2)
Step 5. Solve a problem
The best way to learn how to simplify algebraic fractions is to keep practicing. You can find the solutions right after the problems:
4 (x + 2) (x-13)
(4x + 8) Solution:
(x = 13)
2x2-x
5x Solution:
(2x-1) / 5
Method 3 of 3: Tricks for Complex Problems
Step 1. Find the opposite of the fraction by collecting the negative factors
Suppose you have the equation: 3 (x-4) 5 (4-x) Notice that (x-4) and (4-x) are "almost" identical, but you can't cancel them out because they are one the opposite of the other; however, you can rewrite (x - 4) as -1 * (4 - x), just like you can rewrite (4 + 2x) into 2 * (2 + x). This procedure is called "picking up the negative factor". -1 * 3 (4-x) 5 (4-x) Now you can easily delete the two identical terms (4-x) -1 * 3 (4-x) 5 (4-x) leaving the result - 3/5.
Step 2. Recognize the differences between squares when working with these fractions
In practice, it is a number raised to the square to which another number is subtracted from the power of 2, just like the expression (a2 - b2). The difference between two perfect squares is always simplified by rewriting it as a multiplication between the sum and the difference of the roots; however, you can simplify the difference of perfect squares like this: a2 - b2 = (a + b) (a-b) This is an extremely useful "trick" when looking for similar terms in an algebraic fraction.
Example: x2 - 25 = (x + 5) (x-5).
Step 3. Simplify polynomial expressions
These are complex algebraic expressions, which contain more than two terms, for example x2 + 4x + 3; Fortunately, many of these can be simplified using factoring. The expression described above can be formulated as (x + 3) (x + 1).
Step 4. Remember that you can factor variables as well
This method is especially useful with exponential expressions such as x4 + x2. You can eliminate the major exponent as a factor; in this case: x4 + x2 = x2(x2 + 1).
Advice
- When you collect the factors, check the work done by multiplying, to make sure you find the starting term.
- Try to collect the biggest common factor to completely simplify the equation.
