Rational expressions must be simplified to their minimum factor. This is a fairly simple process if the factor is a single one, but it can be a bit more complex if the factors include multiple terms. Here's what you need to do based on the type of rational expression you need to solve.
Steps
Method 1 of 3: Rational Expression of Monomi
Step 1. Assess the problem
Rational expressions that consist of only monomials are the simplest to reduce. If both terms of the expression each have a term, all you have to do is reduce the numerator and denominator by their greatest common denominator.
- Note that mono means "one" or "single" in this context.
-
Example:
4x / 8x ^ 2
Step 2. Delete the shared variables
Look at the variables that appear in the expression, both in the numerator and in the denominator there is the same letter, you can delete it from the expression respecting the quantities that exist in the two factors.
- In other words, if the variable appears once in the numerator and once in the denominator you can simply delete it since: x / x = 1/1 = 1
- If, on the other hand, the variable appears in both factors but in different quantities, subtract from the one that has a higher power, the one that has the lower power: x ^ 4 / x ^ 2 = x ^ 2/1
-
Example:
x / x ^ 2 = 1 / x
Step 3. Reduce the constants to their lowest terms
If the numerical constants have a common denominator, divide the numerator and denominator by this factor and return the fraction to the minimum form: 8/12 = 2/3
- If the constants of the rational expression do not have a common denominator, it cannot be simplified: 7/5
- If one of the two constants can completely divide the other, it should be considered as a common denominator: 3/6 = 1/2
-
Example:
4/8 = 1/2
Step 4. Write your solution
To determine it, you have to reduce both the variables and the numerical constants and recombine them:
-
Example:
4x / 8x ^ 2 = 1 / 2x
Method 2 of 3: Rational Expressions of Binomials and Polynomials with Monomial Factors
Step 1. Assess the problem
One part of the expression is monomial but the other is binomial or a polynomial. You have to simplify the expression by looking for a monomial factor that can be applied to both the numerator and the denominator.
- In this context, mono means "one" or "single," bi means "two," and poli means "more than two."
-
Example:
(3x) / (3x + 6x ^ 2)
Step 2. Separate the shared variables
If the same variables appear in the numerator and denominator, you can include them in the division factor.
- This is valid only if the variables appear in each term of the expression: x / (x ^ 3 - x ^ 2 + x) = (x) (1) / [(x) (x ^ 2 - x + 1)]
- If a term doesn't contain the variable, you can't use it as a factor: x / x ^ 2 + 1
-
Example:
x / (x + x ^ 2) = [(x) (1)] / [(x) (1 + x)]
Step 3. Separate the shared numeric constants
If the constants in each term of the expression have common factors, divide each constant by the common divisor to reduce the numerator and denominator.
- If one constant divides the other completely, it should be considered as a common divisor: 2 / (2 + 4) = 2 * [1 / (1 + 2)]
- This is valid only if all the terms of the expression share the same divisor: 9 / (6 - 12) = 3 * [3 / (2 - 4)]
- It is not valid if any of the terms of the expression does not share the same divisor: 5 / (7 + 3)
-
Example:
3/(3 + 6) = [(3)(1)] / [(3)(1 + 2)]
Step 4. Bring out the shared values
Combine the variables and reduced constants to determine the common factor. Remove this factor from the expression leaving the variables and constants that cannot be further simplified to each other.
-
Example:
(3x) / (3x + 6x ^ 2) = [(3x) (1)] / [(3x) (1 + 2x)]
Step 5. Write the final solution
To determine this, remove the common factors.
-
Example:
[(3x) (1)] / [(3x) (1 + x)] = 1 / (1 + x)
Method 3 of 3: Rational Expressions of Binomials and Polynomials with Binomial factors
Step 1. Assess the problem
If there are no monomials in the expression, you must report the numerator and denominator to binomial factors.
- In this context, mono means "one" or "single," bi means "two," and poli means "more than two."
-
Example:
(x ^ 2 - 4) / (x ^ 2 - 2x - 8)
Step 2. Break the numerator into binomials
To do this you need to find possible solutions for the variable x.
-
Example:
(x ^ 2 - 4) = (x - 2) * (x + 2).
- To solve for x, you have to put the variable to the left of the equal and the constants to the right of the equal: x ^ 2 = 4.
- Reduce x to single power by taking the square root: √x ^ 2 = √4.
- Remember that the solution of a square root can be both negative and positive. So the possible solutions for x are: - 2, +2.
- Hence the subdivision of (x ^ 2 - 4) in its factors is: (x - 2) * (x + 2).
-
Double check by multiplying the factors together. If you are uncertain about the correctness of your calculations, do this test; you should find the original expression again.
-
Example:
(x - 2) * (x + 2) = x ^ 2 + 2x - 2x - 4 = x ^ 2 - 4
Simplify Rational Expressions Step 12 Step 3. Break the denominator into binomials
To do this you need to determine the possible solutions for x.
-
Example:
(x ^ 2 - 2x - 8) = (x + 2) * (x - 4)
- To solve for x, you have to move the variables to the left of the equal and the constants to the right: x ^ 2 - 2x = 8
- Add to both sides the square root of half the coefficient of x: x ^ 2 - 2x + 1 = 8 + 1
- Simplify both sides: (x - 1) ^ 2 = 9
- Take the square root: x - 1 = ± √9
- Solve for x: x = 1 ± √9
- As with all square equations, x has two possible solutions.
- x = 1 - 3 = -2
- x = 1 + 3 = 4
- Hence the factors of (x ^ 2 - 2x - 8) I'm: (x + 2) * (x - 4)
-
Double check by multiplying the factors together. If you are not sure of your calculations, do this test, you should find the original expression again.
-
Example:
(x + 2) * (x - 4) = x ^ 2 - 4x + 2x - 8 = x ^ 2 - 2x - 8
Simplify Rational Expressions Step 13 Step 4. Eliminate common factors
Determine which binomials, if any, are in common between the numerator and denominator and remove them from the expression. Leave those that cannot be simplified to each other.
-
Example:
[(x - 2) (x + 2)] / [(x + 2) (x - 4)] = (x + 2) * [(x - 2) / (x - 4)]
Simplify Rational Expressions Step 14 Step 5. Write the solution
To do this, remove the common factors from the expression.
-
Example:
(x + 2) * [(x - 2) / (x - 4)] = (x - 2) / (x - 4)
-
-
