© jacoblund/iStock/GettyImages
A rational number can be expressed as a fraction p/q where both p and q are integers and q ≠ 0. To subtract two rational numbers, they must share a common denominator. The same principle applies to rational expressions—polynomial fractions—where the goal is to factor each term to its simplest form before finding a common denominator.
Let’s start with two generic rational numbers: p/q and x/y. To compute p/q − x/y, multiply the first fraction by y/y and the second by q/q (both equal 1). This yields:
\(\frac{p}{q} - \frac{x}{y} = \frac{py}{qy} - \frac{qx}{qy} = \frac{py - qx}{qy}\)
The denominator qy is the least common denominator (LCD). Using the LCD guarantees a correct result and simplifies the expression.
1. Subtract 1/4 from 1/3
Write the subtraction as \(\frac{1}{3} - \frac{1}{4}\). The LCD is 12:
\(\frac{4}{12} - \frac{3}{12} = \frac{1}{12}\)
2. Subtract 3/16 from 7/24
Express the fractions with a common factor of 8:
\(\frac{7}{8\times3} \text{ and } \frac{3}{8\times2}\)
After adjusting, the LCD is 48:
\(\frac{7}{24} - \frac{3}{16} = \frac{14 - 9}{48} = \frac{5}{48}\)
When working with rational expressions, factor both the numerator and the denominator of each term. Cancel any common factors before combining fractions. This reduces the complexity of the LCD and keeps the algebra manageable.
For instance:
\(\frac{x^2 - 2x - 8}{x^2 - 9x + 20} = \frac{(x-4)(x+2)}{(x-5)(x-4)} = \frac{x+2}{x-5}\)
Perform the following subtraction:
\(\frac{2x}{x^2 - 9} - \frac{1}{x + 3}\)
Factor the quadratic in the first denominator:
\(x^2 - 9 = (x+3)(x-3)\)
Rewrite the expression:
\(\frac{2x}{(x+3)(x-3)} - \frac{1}{x+3}\)
The LCD is (x+3)(x-3). Multiply the second fraction by (x-3)/(x-3):
\(\frac{2x - (x-3)}{(x+3)(x-3)} = \frac{x+3}{x^2-9}\)
After simplification, the result is \(\frac{x+3}{x^2-9}\).
