Simplify algebraic fractions

2023 Author: Violet Winter | [email protected]. Last modified: 2023-11-26 19:38

Algebraic fractions look incredibly difficult at first, and for the inexperienced, they can seem daunting to tackle. With a mix of variables, numbers, and even exponents, it's hard to know where to start. Fortunately, however, the same rules that apply to simplifying normal fractions such as 15/25 also apply to algebraic fractions.

method

Method 1 of 3: Simplify Fractions

Step 1. The technical terms for algebraic fractions

The following terms are used in the examples and appear in problems with algebraic fractions:

• Counter:

  The upper number of a fraction (i.e. (x + 5)/ (2x + 3)).

• Denominator:

  The lower number of a fraction (i.e. (x + 5) /(2x + 3)).

• Divider:

  A number whose multiple results in another number. For example, the divisors of 15 are exactly 1, 3, 5, and 15. The divisors of 4 are 1, 2, and 4.

• Common divisor:

  This is a number that divides both the top and bottom of a fraction. For example, in the fraction 3/9, the common factor is 3, since both numbers can be divided by 3.

• Simplified expression:

  This involves eliminating all common factors and grouping like variables (5x + x = 6x) until we have the simplest form of a fraction, equation, or problem. When there is nothing more we can do with the break, it is simplified.

Step 2. Repeat how to simplify simple fractions

These are exactly the same steps that we take to simplify algebraic fractions. Take the example 15/35. To simplify a fraction, we have to find a common factor.

In this case, both numbers can be divided by five so we can remove the 5 from the fraction:

15 → 5 * 3

35 → 5 * 7

Now we can cut out the same coefficients. In this case we can delete the two fives so that we have the simplified result 3/7 obtain.

Step 3. Remove divisors from algebraic expressions as with normal numbers

In the previous example, we could easily remove the 5 from 15, and the same goes for more complex expressions like 15x - 5. Find a factor that both numbers have in common. Here the result is 5, because you can divide both 15x and -5 by the number five. As before, we remove the common factor and multiply it by what's left.

15x - 5 = 5 * (3x - 1) To check your work, just multiply the five again by the new expression (in the numerator and in the denominator) - you will end up with the same numbers that you started with.

Step 4. You can remove complex terms as well as simple ones

The same principle as with simple fractions also applies to algebraic fractions. This is the easiest way to simplify fractions while you work on them. Let's look at the fraction (x + 2) (x-3) (x + 2) (x + 10). Note that the term (x + 2) occurs both in the numerator (above) and in the denominator (below). This way you can remove it to simplify the algebraic fraction, just like you removed the 5 from 15/35: (x + 2) (x-3) → (x-3) (x + 2) (x + 10) → (x + 10) So we have our final result: (x-3) / (x + 10)

Method 2 of 3: Simplify Algebraic Fractions

Step 1. Find common factors in the numerator or the upper part of the fraction

The first thing to do when trying to simplify an algebraic fraction is to simplify each part of the fraction. Start with the top part and remove as many dividers as you can. In this section we use the example:

9x-3

15x + 6

Start with the numerator: 9x - 3. There is a common divisor of 9x and -3: 3. Factor 3 as with a normal number so that we get 3 * (3x-1). This is our new counter:

3 (3x-1)

15x + 6

Step 2. Find common factors in the denominator

To continue the example above, let's look at the denominator, 15x + 6. Here, too, we're looking for a number that can divide both parts. Again, we can factor out the 3 so that we get 3 * (5x + 2). We write our new denominator as:

3 (3x-1)

3 (5x + 2)

Step 3. Remove the same terms

This is the stage where we really simplify the fraction. Take all terms that appear in both the numerator and denominator and remove them. In this case we can remove the 3 from both the top and the bottom.

3 (3x-1) → (3x-1)

3 (5x + 2) → (5x + 2)

Step 4. When is the expression completely simplified?

A fraction is simplified when there are no other common factors above and below. Remember that no factors can be removed from inside the brackets - in the example, you cannot factor out the x of 3x and 5x, since the complete terms are actually (3x -1) and (5x + 2). Thus the example is completely simplified and we get the final result:

(3x-1)

(5x + 2)

Step 5. Try an exercise

The best way to learn is to keep trying to simplify algebraic fractions. The results are under the tasks.

4 (x + 2) (x-13)

(4x + 8) Result:

(x = 13)

2x²-x

5x Answer:

(2x-1) / 5

Method 3 of 3: Tricks for Difficult Tasks

Step 1. “Reverse the sign” in some terms of the fraction by factoring out negative numbers

Suppose we have the fraction:

3 (x-4)

5 (4-x)

Note that (x- 4) and (4-x) are almost identical, but they cannot be truncated because they have opposite signs. However, (x - 4) can be written as -1 * (4 - x), in the same way that (4 + 2x) can be written as 2 * (2 + x). This is called "sign factoring".

-1 * 3 (4-x)

5 (4-x)

Now we can easily remove the two identical (4-x):

-1 * 3 (4-x)

5 (4-x)

and get our final result - 3/5

Step 2. As you work, see the difference of two squares

The difference of two squares is simply a square number subtracted from another, like the expression (a² - b²). The difference of squares can always be simplified to:

a² - b² = (a + b) (a-b) This can be incredibly helpful when looking for like terms in algebraic fractions.

• Example: x² - 25 = (x + 5) (x-5)

Step 3. Simplify all polynomial expressions

Polynomials are complex algebraic expressions with more than two terms, like x² + 4x + 3. Fortunately, many polynomials can be simplified by polynomial factorization. For example, the above term can be rewritten as (x + 3) (x + 1).

Step 4. Note that variables can also be factored out

This is especially useful in expressions with exponents, such as x⁴ + x². You can factor out the variable with the greatest common exponent. In this case x is⁴ + x² = x²(x² + 1).

Tips

• Always factor out the largest possible numbers to completely simplify your fraction.
• Check your work if you factor out by multiplying again - you should get the same numbers again as at the beginning.