Algebraic fractions seem very difficult when first seen and can be intimidating to students who are not very proficient in them. With a mix of variables, numbers, and even exponents, it's hard to know where to start. Luckily, the same rules you need to simplify regular fractions, like 15/25, also apply to algebraic fractions.
Steps
Method 1 of 3: Simplify Fractions
Step 1. Know the vocabulary for algebraic fractions
The following terms are used in the examples and are very common in problems where there are algebraic fractions:
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Numerator:
the top of the fraction (for example (x + 5)/ (2x + 3)).
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Denominator:
the bottom of the fraction (for example (x + 5) /(2x + 3)).
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Common denominator:
This is the number that you can divide between the top and bottom numbers. For example, in the fraction 3/9, the common denominator is 3, since both numbers are divisible by 3.
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Factor:
a number that is multiplied to have another. For example, the factors of 15 are 1, 3, 5, and 15. The factors of 4 are 1, 2, and 4.
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Simplified equation:
It consists of eliminating all the common factors and grouping similar variables (5x + x = 6x) until you have the most basic form of a fraction, equation or problem. If you can't do anything else with the fraction, it's already simplified.
Step 2. Review how to solve simple fractions
These are the same steps you must follow to solve algebraic fractions. Take as an example: 15/35. In order to simplify a fraction, you must find the common denominator. In this case, both numbers can be divided by 5, so you can remove 5 from the fraction:
15 → 5 * 3
35 → 5 * 7
Now you can cancel the terms similar. In this case, you can cancel the two fives, leaving the simplified answer of 3/7.
Step 3. Eliminate factors from algebraic expressions as if they were normal numbers
In the example above, you can eliminate the 5 from 15 and the same principle applies to more complex expressions like, 15x - 5. Find a factor that both numbers have in common. For example here, the answer is 5, since you can divide 15x and -5 by 5. As in the previous examples, remove the common factor and multiply it by what is left.
15x - 5 = 5 * (3x - 1) To check your work, simply multiply the 5 in your expression: you will end up with the same number you started with.
Step 4. Understand that you can also cancel more complex terms as well as simpler ones
The same principle of common fractions is used in algebraic fractions. This is the easiest way to simplify fractions. For instance:
(x + 2) (x-3)
(x + 2) (x + 10)
Notice how the term (x + 2) is common in the numerator (top) and the denominator (bottom). So you can eliminate that term to simplify the algebraic fraction, just as you eliminated the 5 from 15/35:
(x + 2) (x-3) → (x-3)
(x + 2) (x + 10) → (x + 10) Therefore the final answer is: (x-3) / (x + 10)
Method 2 of 3: Simplify Algebraic Fractions
Step 1. Find a common factor in the numerator or at the top of the fraction
The first thing you should do when simplifying an algebraic fraction is to simplify each part of the fraction. Start at the top, factoring as many numbers as you can. For example, pay attention to the following equation:
9x-3
15x + 6
Start with the numerator: 9x - 3. There is a common factor in both 9x and -3: 3. Factor 3 as you would any other number, which gives 3 * (3x-1). This is the new numerator:
3 (3x-1)
15x + 6
Step 2. Find a common factor in the denominator
Following the previous example, isolate the denominator, 15x + 6. Again, find a number that you can divide both parts into. Here you can use the 3 again, which gives 3 * (5x +2). Write the new denominator:
3 (3x-1)
3 (5x + 2)
Step 3. Remove the terms
This is when you should simplify the fraction. Take the terms that are in both the numerator and the denominator and eliminate them. In this case, you can remove the top 3 and bottom 3.
3 (3x-1) → (3x-1)
3 (5x + 2) → (5x + 2)
Step 4. Understand when you can't completely simplify an equation
A fraction is simplified when there are no more common factors, neither above nor below. Remember that the factors inside the parentheses cannot be eliminated. In the example problem, you cannot factor the x of the 3x and 5x, since the complete terms are (3x -1) and (5x + 2). Therefore, the example is already simplified, which makes the final answer The next:
(3x-1)
(5x + 2)
Step 5. Do a practice problem
The best way to learn is to keep practicing and simplifying algebraic fractions. The answers are below the problems.
4 (x + 2) (x-13)
(4x + 8) Answer:
(x = 13)
2x2-x
5x Answer:
(2x-1) / 5
Method 3 of 3: Tricks for Tough Problems
Step 1. Invert parts of the fraction, factoring out the negative numbers
For example, with the following equation:
3 (x-4)
5 (4-x)
Notice how (x-4) and (4-x) are almost identical, but they cannot be eliminated because they are reversed. However, (x - 4) can be written as follows: -1 * (4 - x), in the same way that you can write (4 + 2x) as 2 * (2 + x). This is called "factoring the negative."
-1 * 3 (4-x)
5 (4-x)
Now you can remove the two (4-x):
-1 * 3 (4-x)
5 (4-x)
Which leaves you with the final answer of - 3/5.
Step 2. Recognize the difference of the squares as you work
The difference between two squares is simply subtracting square number from another, as in (a2 - b2). The difference of perfect squares is always simplified into two parts, adding and subtracting the square roots. You can always simplify the difference of the perfect squares as follows:
to2 - b2 = (a + b) (a-b) This is very useful when trying to find like terms in algebraic fractions.
- Example: x2 - 25 = (x + 5) (x-5)
Step 3. Simplify the polynomial expressions
Polynomials are very complex algebraic expressions with more than two terms, such as x2 + 4x + 3. Fortunately, many polynomials can be simplified by factoring polynomials. The previous expression, for example, can be written as follows: (x + 3) (x + 1).
Step 4. Remember that variables can also be factored
This is very useful in expressions with exponents, like x4 + x2. You can eliminate the largest exponent as a factor. In this case it would be x4 + x2 = x2(x2 + 1).
Advice
- Always factor the largest numbers you can to keep the equation as simple as possible.
- Check your work when factoring by multiplying the factor in the equation. You will get the same number you started with.