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If there are more than two rational expressions, we find their difference by rewriting all expressions in equivalent forms using the LCD (least common denominator), so they have the same denominator. Then we subtract numerator from left to right, remembering to correctly distribute the "minus" sign.
To find the difference between more than two rational expressions, you can follow these steps:
1. Identify the rational expressions you want to find the difference between. Let's say you have three rational expressions: expression1, expression2, and expression3.
2. Rewrite each rational expression in equivalent forms with a common denominator. To do this, find the least common denominator (LCD) of the denominators of all the expressions. This will be the shared denominator that you will use for all the expressions.
3. Rewrite each expression with the LCD as the denominator. Multiply both the numerator and denominator of each expression by the appropriate factors to make the denominators match the LCD.
4. Subtract the expressions from left to right, remembering to properly handle the "minus" sign. To do this, subtract the numerators of the expressions while keeping the common denominator. The result will be a single rational expression.
Here's the step-by-step process using the example of three rational expressions:
Expression1: \( \frac{a}{b} \)
Expression2: \( \frac{c}{d} \)
Expression3: \( \frac{e}{f} \)
1. Identify the expressions: expression1, expression2, expression3.
2. Find the LCD: This will be the least common multiple of the denominators b, d, and f.
3. Rewrite each expression with the LCD as the denominator:
Expression1: \( \frac{a}{b} \) becomes \( \frac{a \cdot (d \cdot f)}{b \cdot (d \cdot f)} = \frac{a \cdot df}{bdf} \)
Expression2: \( \frac{c}{d} \) becomes \( \frac{c \cdot (b \cdot f)}{d \cdot (b \cdot f)} = \frac{c \cdot bf}{bdf} \)
Expression3: \( \frac{e}{f} \) becomes \( \frac{e \cdot (b \cdot d)}{f \cdot (b \cdot d)} = \frac{e \cdot bd}{bdf} \)
4. Subtract the expressions:
\( \frac{a \cdot df}{bdf} - \frac{c \cdot bf}{bdf} - \frac{e \cdot bd}{bdf} = \frac{a \cdot df - c \cdot bf - e \cdot bd}{bdf} \)
The result is a single rational expression with the common denominator bdf.
To find the difference between more than two rational expressions, you can follow these steps:
1. Rewrite all the expressions in equivalent forms with a common denominator.
- Identify the least common multiple (LCM) of the denominators of the rational expressions.
- Multiply each expression's numerator and denominator by the necessary factors to make the denominators the same as the LCM.
2. Once all expressions have the same denominator, subtract them from left to right.
- Start by subtracting the first expression from the second expression.
- Then subtract the next expression from the previous result, and continue until all expressions have been subtracted.
3. Remember to correctly distribute the negative sign.
- When subtracting a rational expression, distribute the negative sign to each term in the numerator after subtracting.
By following these steps, you can find the difference between more than two rational expressions. Remember to simplify the resulting expression if possible.