Adding and Subtracting: Rational expressions
Adding and subtracting rational expressions involves combining fractions that consist of polynomials in both the numerator and denominator. A crucial aspect of working with these expressions is ensuring that they have a common denominator, similar to standard fractions. To achieve this, one must identify the least common denominator (LCD) by factoring the denominators of the rational expressions involved.
Once the LCD is determined, each rational expression is adjusted by multiplying by any missing factors needed to match the LCD. After achieving a common denominator, the numerators can be combined through addition or subtraction, while being mindful to distribute negative signs appropriately.
Finally, any coefficients of like terms in the numerator are combined, leading to a simplified expression, if possible. It is essential to ensure that the resulting rational expression retains a non-zero denominator, as this defines its domain and keeps the expression valid. Understanding these steps is foundational for working effectively with rational expressions in algebra.
Adding and Subtracting: Rational expressions
A rational expression is the quotient of two polynomials. In other words, a rational expression is a fraction with a polynomial numerator and a polynomial denominator. Below is an example of a rational expression.
Similar to a fraction, a rational expression needs to have a non-zero denominator because if its denominator is zero, it will become undefined.
Domain of Rational Expressions
The domain of a rational expression is the set of real numbers that provide a defined value for the rational expression. Because a rational expression becomes undefined when its denominator is zero, all real numbers that result in a denominator with a value of zero are eliminated from the domain of the rational expression.
An example of a rational expression with a denominator that assumes a value of zero for some values of x can be seen below.
Factoring a rational expression will make it easier to identify the values of x that result in a denominator with a value of zero. The rational expression above can be factored out as follows:
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With the simplified rational expression, it is easier to see that the denominator becomes zero if the value of x is either −2 or −3. Thus, the domain of the function above is all real numbers except x = −2 or −3.
Simplifying Rational Expressions
As with fractions consisting of integers, rational expressions are reduced to the lowest terms (or simplified) by completely factoring out the numerator and the denominator. All the common factors of the numerator and denominator are then cancelled out (or eliminated).
Below is an example of a rational expression that needs to be simplified.
First, completely factor out both the numerator and the denominator. This will result in the expression below.
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The common factors of the numerator and the denominator are x – 3, x + 1, and x + 6. Cancel these out to get the simplified rational expression:
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Adding and Subtracting Rational Expressions
Similar to adding or subtracting fractions, rational expressions need to have the same denominator values so that their numerators can be added or subtracted. Below is an example problem.
Since the denominators are not the same, factor them out completely to determine the least common denominator of the three rational expressions.
From the factored out denominators, it can be determined that the least common denominator of the three rational expressions is x(x + 1)(x + 2). The next step is to multiply the rational expressions by the missing factors to obtain the least common denominator for all three of them.
The next step is to expand all three numerators to obtain:
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Now that the numerators have been expanded and the least common denominator has been obtained, the rational expressions can be combined. The actual subtraction and addition can now be performed. (Remember to distribute the subtraction or negative sign to all terms of the numerator.)
Combine like terms by adding or subtracting the coefficients of the variables (x) with the same exponents to obtain:
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Since the numerator cannot be simplified any further, the final answer is
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Bibliography
Aufmann, Richard, Vernon Barker, and Richard Nation. College Algebra and Trigonometry. 7th ed. Belmont, CA: Cengage, 2011. Print.
McKellar, Danica. Hot X: Algebra Exposed! New York: Plume, 2011.
Young, Cynthia Y. Algebra and Trigonometry. 3rd ed. Hoboken, NJ: Wiley, 2013. Print.