How to Simplify Rational Expressions
Three Methods:Monomial Rational ExpressionsBinomial or Polynomial Rational Expressions with Monomial FactorsBinomial or Polynomial Rational Expressions with Binomial Factors
Rational expressions need to be simplified to their lowest like factors. This is a fairly simple process if the like factor is a single-term factor, but it can be a little more detailed when the factor includes multiple terms. Here's what you need to do depending on the type of rational expression you're faced with.
Steps
Method 1 Monomial Rational Expressions
- 1Examine the problem.^{[1]} Rational expressions only consisting of monomials are the the easiest to simplify. If both terms in the expression only have one term apiece, all you need to do is to directly reduce the numerator and denominator to their lowest shared terms.
- Note that mono means “one” or “single” in this context.
- Example: 4x/8x^2
- 2Cancel out any shared variables. Look at the letter variables in the expression. If the same variable appears in both the numerator and denominator, you can cancel this variable out in the amount that it appears in for both parts of the expression.
- In other words, if the the variable only appears in the expression once in the numerator and once in the denominator, the variable can be completely canceled out: x/x = 1/1 = 1
- If, however, the variable appears more times in either the numerator or the denominator, but appears at least once in the other portion of the expression, subtract the power that variable is raised to in the smaller portion of the expression from the power the variable is raised to in the larger portion: x^4/x^2 = x^2/1
- Example: x/x^2 = 1/x
- 3Reduce the constants to lowest terms. If the numerical constants have like factors, divide the constant in the numerator and the constant in the denominator by that like factor to reduce the fraction to its smallest form: 8/12 = 2/3
- If the constants in a rational expression do not share like factors, they cannot be simplified: 7/5
- If one constant divides into the other evenly, that is considered to be the like factor: 3/6 = 1/2
- Example: 4/8 = 1/2
- 4Write your final answer. To determine your final answer, you must combine the reduced variables and the reduced constants together again.^{[2]}
- Example: 4x/8x^2 = 1/2x
Method 2 Binomial or Polynomial Rational Expressions with Monomial Factors
- 1Examine the problem. If one portion of the rational expression is a monomial but the other is a binomial or polynomial, you may need to simplify the expression by determining a monomial factor that can be applied to both the numerator and denominator.
- In this context, mono means “one” or “single,” bi means “two,” and poly means “multiple.”
- Example: (3x)/(3x + 6x^2)
- 2Separate any shared variables. If any letter variable appears in all terms within the equation, you can include that as part of the factored out term.
- This only works if the variable appears in each term of the equation: x/x^3 – x^2 + x = (x)(x^2 – x + 1)
- If one term of the equation does not contain the variable, you cannot factor it out: x/x^2 + 1
- Example: x/(x + x^2) = [(x)(1)] / [(x)(1 + x)]
- 3Separate any shared constants. If the numerical constants in each term have like factors, divide each constant in those terms by that like factor to reduce the numerator and denominator.
- If one constant divides into the other evenly, that is considered to be the like factor: 2 / (2 + 4) = 2 * [1 / (1 + 2)]
- Note that this only works if all terms in the expression share at least one like factor: 9 / (6 – 12) = 3 * [3 / (2 – 4)]
- This will not work if any of the terms in the expression do not share the same factor: 5 / (7 + 3)
- Example: 3/(3 + 6) = [(3)(1)] / [(3)(1 + 2)]
- 4Factor out the shared elements. Combine the reduced variables and the reduced constants together again to determine the common factor. Remove this factor from the expression, leaving behind the variables and constants that are not shared among all the terms.
- Example: (3x)/(3x + 6x^2) = [(3x)(1)] / [(3x)(1 + 2x)]
- 5Write your final answer. To determine the final answer, remove the shared factor from the expression altogether.
- Example: [(3x)(1)] / [(3x)(1 + 2x)] = 1/(1 + 2x)
Method 3 Binomial or Polynomial Rational Expressions with Binomial Factors
- 1Examine the problem. If there are no monomial terms in the rational expression, you will need to separate the numerator and denominator into binomial factors.
- In this context, mono means “one” or “single,” bi means “two,” and poly means “multiple.”
- Example: (x^2 - 4) / (x^2 - 2x - 8)
- 2Separate the numerator into binomial factors. In order to separate the numerator into its factors, you need to determine the possible solutions for your variable, x.
- Example: (x^2 – 4) = (x - 2) * (x + 2)
- In order to solve for x, you need to move the constant to one side and keep the variable on the other side: x^2 = 4
- Reduce the x to a single power by taking the square root of both sides: √x^2 = √4
- Remember that the square root of any number can be positive or negative. As such, the possible answers for x are: -2, +2
- As such, when breaking (x^2 – 4) into its factors, the factors are: (x - 2) * (x + 2)
- Double-check your factors by multiplying them together. If you are uncertain about whether or not you factored this part of the rational expression correctly, you can multiply these factors together to make sure that they equal the original expression. Remember to use FOIL when appropriate: first, outside, inside, last.
- Example: (x - 2) * (x + 2) = x^2 + 2x - 2x – 4 = x^2 – 4
- Example: (x^2 – 4) = (x - 2) * (x + 2)
- 3Separate the denominator into binomial factors. In order to separate the numerator into its factors, you need to determine the possible solutions for your variable, x.
- Example: (x^2 - 2x – 8) = (x + 2) * (x – 4)
- In order to solve for x, you need to move the constant to one side and keep all terms including variables on the other side: x^2 − 2x = 8
- Square half of the x-term coefficient and add the value to both sides: x^2 − 2x + 1 = 8 + 1
- Simplify the right side and write the perfect square on the right: (x − 1)^2 = 9
- Take the square root of both sides: x − 1 = ±√9
- Solve for x: x = 1 ±√9
- As with other quadratic equations, x will have two possible solutions.^{[3]}
- x = 1 - 3 = -2
- x = 1 + 3 = 4
- As such, (x^2 - 2x – 8) factors into (x + 2) * (x – 4)
- Double-check your factors by multiplying them together. If you are uncertain about whether or not you factored this part of the rational expression correctly, you can multiply these factors together to make sure that they equal the original expression. Remember to use FOIL when appropriate: first, outside, inside, last.
- Example: (x + 2) * (x – 4) = x^2 – 4x + 2x – 8 = x^2 - 2x - 8
- Example: (x^2 - 2x – 8) = (x + 2) * (x – 4)
- 4Cut away the common factors. Determine which binomial factor, if any, is common between the numerator and denominator. Remove this factor from the expression, leaving behind the binomial factors that are not alike.
- Example: [(x - 2)(x + 2)] / [(x + 2)(x – 4)] = (x + 2) * [(x – 2) / (x – 4)]
- 5Write your final answer. To determine the final answer, remove the shared factor from the expression altogether.^{[4]}
- Example: (x + 2) * [(x – 2) / (x – 4)] = (x – 2) / (x – 4)
Things You'll Need
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Categories: Algebra