Rational expressions are expressions in the form of a fraction (or ratio) of two polynomials. As with regular fractions, a rational expression can be simplified. This is a very simple process if the factor is a monomial (or single-term factor); however, it can be a bit more detailed if the factor includes multiple terms.
Steps
Method 1 of 3: Factoring Monomials
Step 1. Analyze the expression
To use this method you must place a monomial in the numerator and denominator of your rational expression. A monomial is a polynomial with one term.
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For example, the expression 4x16x2 { displaystyle { frac {4x} {16x ^ {2}}}}
tiene un término en el numerador y otro término en el denominador. Por lo tanto, cada uno es un monomio.
- La expresión 4x+416x2−2{displaystyle {frac {4x+4}{16x^{2}-2}}}
tiene dos binomios y, por lo tanto, no se puede resolver mediante este método.
Step 2. Factor the numerator
To do this, write down the factors that you would multiply to get the monomial, including the variable. For more information on how to factor, read this article on how to factor a number. Rewrite the expression using the factors in the numerator and denominator.
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For example, 4x { displaystyle 4x}
lo factorizarías como 2×2×x{displaystyle 2\times 2\times x}
y 16x2{displaystyle 16x^{2}}
como 2×2×2×2×x×x{displaystyle 2\times 2\times 2\times 2\times x\times x}
. Por lo tanto, luego de factorizar la expresión se vería así:
2×2×x2×2×2×2×x×x{displaystyle {frac {2\times 2\times x}{2\times 2\times 2\times 2\times x\times x}}}
Step 3. Cancel the factors that share the numerator and denominator
To do this, cross out the matching factors. Cancellation occurs because you divide a factor by itself, which is equal to 1.
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For example, in the numerator and denominator you can cross out two "2" and one "x" and the expression would look like this:
2 × 2 × x2 × 2 × 2 × 2 × x × x { displaystyle { frac {{ cancel {2}} times { cancel {2}} times { cancel {x}}} {{ cancel {2}} times { cancel {2}} times 2 \ times 2 \ times { cancel {x}} times x}}}
Step 4. Rewrite the expression with the remaining factors
Remember that the canceling terms give 1 as a result. Therefore, if you have canceled all the terms in the numerator or denominator, you will still have “1” left.
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For instance:
2 × 2 × x2 × 2 × 2 × 2 × x × x { displaystyle { frac {{ cancel {2}} times { cancel {2}} times { cancel {x}}} {{ cancel {2}} times { cancel {2}} times 2 \ times 2 \ times { cancel {x}} times x}}}
12×2×x{displaystyle {frac {1}{2\times 2\times x}}}
Step 5. Finish any multiplication in the numerator or denominator
This way you will obtain the rational, simplified and final expression.
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For instance:
12 × 2 × x { displaystyle { frac {1} {2 \ times 2 \ times x}}}
14x{displaystyle {frac {1}{4x}}}
Método 2 de 3: Factorizar factores comunes monomios
Step 1. Analyze the rational expression
To use this method you must place at least one binomial in your expression. It can be in the numerator, denominator, or both. A binomial is a polynomial with two terms.
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For example, the expression 4x16x2−2x { displaystyle { frac {4x} {16x ^ {2} -2x}}}
tiene dos términos en el denominador. Por lo tanto, este contiene un binomio.
Step 2. Find a monomial factor that is common to the numerator and denominator
The factor must be common to all terms in the expression. Factor that term and rewrite it in the expression.
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For example, the monomial 2x { displaystyle 2x}
es común a cada término de la expresión 4x16x2−2x{displaystyle {frac {4x}{16x^{2}-2x}}}
. Por lo tanto, después de factorizar dicho término del numerador y denominador, la expresión se verá así:
2x(2)2x(8x−1){displaystyle {frac {2x(2)}{2x(8x-1)}}}
Step 3. Cancel the common factor
When factoring the monomials of the numerator and denominator you get 1 since you will be dividing said term by itself.
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For instance:
2x (2) 2x (8x − 1) { displaystyle { frac {2x (2)} {2x (8x-1)}}}
2x(2)2x(8x−1){displaystyle {frac {{cancel {2x}}(2)}{{cancel {2x}}(8x-1)}}}
Step 4. Rewrite the expression after canceling the monomial
Thus you will obtain the simplified rational expression. If you factor it correctly, there will be no factors that are common to each term in the numerator and denominator.
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For instance:
2x (2) 2x (8x − 1) { displaystyle { frac {{ cancel {2x}} (2)} {{ cancel {2x}} (8x-1)}}}
28x−1{displaystyle {frac {2}{8x-1}}}
Método 3 de 3: Factorizar factores comunes binomios
Step 1. Analyze the expression
This method works with expressions that have second-degree polynomials in the numerator and denominator. A second-degree polynomial is one that has a term raised to the power of 2.
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For example, the expression x2−4x2−2x − 8 { displaystyle { frac {x ^ {2} -4} {x ^ {2} -2x-8}}}
tiene un polinomio de segundo grado en el numerador y denominador; por lo tanto, puedes utilizar este método para simplificarlo.
Step 2. Factor the polynomial of the numerator into two binomials
Here what you will do is look for two binomials that give you the original polynomial when multiplying them by the FOIL method. For more information on how to factor a second degree polynomial, read this article. Rewrite the expression with the factored numerator.
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For example, x2−4 { displaystyle x ^ {2} -4}
se puede factorizar como (x−2)(x+2){displaystyle (x-2)(x+2)}
. Por lo tanto, ahora la expresión se vería así:
(x−2)(x+2)x2−2x−8{displaystyle {frac {(x-2)(x+2)}{x^{2}-2x-8}}}
Step 3. Factor the polynomial in the denominator into two binomials
Once again, what you will do is find two binomials that give you the original polynomial when multiplying them. Rewrite the expression with the denominator factored.
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For example, x2−2x − 8 { displaystyle x ^ {2} -2x-8}
se puede factorizar como (x+2)(x−4){displaystyle (x+2)(x-4)}
. Por lo tanto, ahora la expresión se vería así:
(x−2)(x+2)(x+2)(x−4){displaystyle {frac {(x-2)(x+2)}{(x+2)(x-4)}}}
Step 4. Cancel the binomial factors that are common for the numerator and denominator
A binomial factor is an expression in parentheses. You can factor them since dividing a factor by itself you get 1.
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For instance:
(x − 2) (x + 2) (x + 2) (x − 4) { displaystyle { frac {(x-2) (x + 2)} {(x + 2) (x-4)} }}
(x−2)(x+2)(x+2)(x−4){displaystyle {frac {(x-2){cancel {(x+2)}}}{{cancel {(x+2)}}(x-4)}}}
Step 5. Rewrite the expression with the remaining factors
Remember that you will have 1 if you cancel all the factors. Thus you will obtain the simplified and final expression.
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For instance:
(x − 2) (x + 2) (x + 2) (x − 4) { displaystyle { frac {(x-2) { cancel {(x + 2)}}} {{ cancel {(x + 2)}} (x-4)}}}
x−2x−4{displaystyle {frac {x-2}{x-4}}}