Content

• Steps
• Method 1 of 3: Reducing Fractions
• Method 2 of 3: Reducing Algebraic Fractions
• Method 3 of 3: Special Techniques
• Tips
• Warnings

At first glance, algebraic fractions seem very complex, and an untrained student may think that nothing can be done with them. The jumble of variables, numbers and even degrees inspires fear. However, the same rules are used to reduce common (eg 15/25) and algebraic fractions.

Steps

Method 1 of 3: Reducing Fractions

1. 1 Master the terms used to describe algebraic fractions. The terms below are common when considering algebraic fractions, and they will be used further when considering examples:
   • Numerator... The upper part of the fraction (for example, (x + 5)/ (2x + 3)).
   • Denominator... The lower part of the fraction (for example, (x + 5) /(2x + 3)).
   • Common divisor... This is the name of the number by which the upper and lower parts of the fraction are divided. For example, 3/9 has a common factor of 3, since both are divisible by 3.
   • Factor... These are numbers that, when multiplied, produce a given number. For example, 15 can be expanded into factors of 1, 3, 5, and 15. The factors of 4 are 1, 2, and 4.
   • Simplified form... To get a simplified form of an algebraic fraction, cancel all common factors and group the same variables (for example, 5x + x = 6x). If nothing else is canceled, then the fraction has a simplified form.
2. 2 Check out the steps for simple fractions. Operations with ordinary and algebraic fractions are similar. For example, let's take the fraction 15/35. To simplify this fraction, one should find common divisor... Both numbers are divisible by five, so we can highlight 5 in both the numerator and denominator: 15 → 5 * 335 → 5 * 7 Now you can reduce common factors, that is, cross out 5 in the numerator and denominator. As a result, we get a simplified fraction 3/7.
3. 3 In algebraic expressions, common factors are distinguished in the same way as in ordinary ones. In the previous example, we were able to easily distinguish 5 out of 15 - the same principle applies to more complex expressions such as 15x - 5. Find the common factor. In this case, it will be 5, since both terms (15x and -5) are divisible by 5. As before, select the common factor and carry it over to the left.15x - 5 = 5 * (3x - 1) To check if everything is correct, it is enough to multiply the expression in the brackets by 5 - the result will be the same numbers as at the beginning.
4. 4 Complex members can be selected in the same way as simple ones. For algebraic fractions, the same principles apply as for ordinary ones. This is the easiest way to reduce a fraction. Consider the following fraction: (x + 2) (x-3)(x + 2) (x + 10) Note that both the numerator (above) and the denominator (below) contain the term (x + 2), so it can be canceled in the same way as the common factor 5 in the fraction 15/35 : (x + 2)(x-3) → (x-3)(x + 2)(x + 10) → (x + 10) As a result, we get a simplified expression: (x-3) / (x + 10)

Method 2 of 3: Reducing Algebraic Fractions

1. 1 Find the common factor in the numerator, that is, at the top of the fraction. When canceling an algebraic fraction, the first step is to simplify both parts of it. Start with the numerator and try to expand it into as many factors as possible. Consider the following fraction in this section: 9x-315x + 6 Let's start with the numerator: 9x - 3. For 9x and -3, the common factor is 3. Move 3 out of the parentheses, as is done with ordinary numbers: 3 * (3x-1). As a result of this transformation, the following fraction will be obtained: 3 (3x-1)15x + 6
2. 2 Find the common factor in the numerator. Let's continue with the above example and write out the denominator: 15x + 6. As before, find the number by which both parts are divisible. And in this case, the common factor is 3, so you can write: 3 * (5x +2). Let's rewrite the fraction as follows: 3 (3x-1)3 (5x + 2)
3. 3 Reduce identical members. At this step, you can simplify the fraction. Cancel the identical terms in the numerator and denominator. In our example, this number is 3.
   3(3x-1) → (3x-1)
   3(5x + 2) → (5x + 2)
4. 4 Determine that the fraction is of the simplest form. The fraction is completely simplified when there are no common factors left in the numerator and denominator. Note that you cannot cancel out those terms that are inside the brackets - in the given example, there is no way to separate x from 3x and 5x, since the full terms are (3x -1) and (5x + 2). Thus, the fraction defies further simplification, and the final answer looks like this:
   (3x-1)
   (5x + 2)
5. 5 Practice cutting fractions yourself. The best way to learn the method is to solve problems on your own. The correct answers are given below the examples. 4 (x + 2) (x-13)(4x + 8) Answer: (x = 13) 2x-x5x Answer:(2x-1) / 5

Method 3 of 3: Special Techniques

1. 1 Move the negative sign outside the fraction. Suppose the following fraction is given: 3 (x-4)5 (4-x) Note that (x-4) and (4-x) are “almost” identical, but they cannot be shortened right away as they are “upside down”. However, (x - 4) can be written as -1 * (4 - x), just as (4 + 2x) can be written as 2 * (2 + x). This is called “reversal of sign”. -1 * 3 (4-x)5 (4-x) Now you can cancel the same terms (4-x): -1 * 3(4-x)5(4-x) So, we get the final answer: -3/5.
2. 2 Learn to recognize the difference in squares. Difference of squares is when the square of one number is subtracted from the square of another number, as in the expression (a - b). The difference of complete squares can always be decomposed into two parts - the sum and the difference of the corresponding square roots. Then the expression will take the following form: a - b = (a + b) (a-b) This technique is very useful when looking for common terms in algebraic fractions.
   • Example: x - 25 = (x + 5) (x-5)
3. 3 Simplify polynomial expressions. Polynomials are complex algebraic expressions that have more than two terms, such as x + 4x + 3. Fortunately, many polynomials can be factorized. For example, the above expression can be written as (x + 3) (x + 1).
4. 4 Remember that variables can also be factorized. This is especially useful in the case of exponential expressions such as x + x. Here you can place the variable outside the brackets to a lesser extent. In this case, we have: x + x = x (x + 1).

Tips

• Check if you have factorized this or that expression correctly. To do this, multiply the factors - the result should be the same expression.
• To completely simplify a fraction, always select the largest factors.

Warnings

• Never forget about the properties of exponents! Try to remember these properties firmly.