explain how to add rational expressions
same as you do with ordinary numeric fractions. Find a common denominator and use that to combine them.
(x-2)/(x+3) + (2x-7)/(x-5) - 3/(x-2)
Common denominator is (x+3)(x-5)(x-2). So, now you can multiply each term by the needed factors:
(x-2)(x-5)(x-2) + (2x-7)(x+3)(x-2) - 3(x+3)(x-5)
-------------------------------------------------------------
(x+3)(x-5)(x-2)
3x^3-17x^2+11x+67
----------------------------
(x+3)(x-5)(x-2)
Thank you so much!
To add rational expressions, follow these steps:
1. Simplify each rational expression separately if possible. Simplify by factoring, canceling common factors, and reducing to lowest terms.
2. Find a common denominator for the rational expressions. To do this, factor each denominator completely and identify all the unique factors. Then, multiply these unique factors together to get the common denominator.
3. Rewrite each rational expression using the common denominator. Multiply the numerator and denominator of each expression by the factors of the common denominator that are not already present in the denominator of that expression.
4. Add the rewritten rational expressions together. Combine the numerators over the common denominator to obtain a single rational expression.
5. If possible, simplify the resulting expression by factoring and canceling common factors.
Let's go through an example to illustrate these steps:
Example:
Add the rational expressions 3/(x-1) + 2/(x+2)
Solution:
Step 1: Both expressions are already in their simplest form, so we move to Step 2.
Step 2: The common denominator is (x-1)(x+2) because it includes both denominators.
Step 3: Rewrite the rational expressions using the common denominator:
3/(x-1) = 3(x+2)/[(x-1)(x+2)]
2/(x+2) = 2(x-1)/[(x-1)(x+2)]
Step 4: Add the rewritten rational expressions together:
3(x+2)/[(x-1)(x+2)] + 2(x-1)/[(x-1)(x+2)]
Combining the numerators, we get:
(3x + 6 + 2x - 2)/[(x-1)(x+2)]
This simplifies to:
(5x + 4)/[(x-1)(x+2)]
Step 5: If necessary, simplify the resulting expression further. In this case, it is already in its simplest form, so we don't need to simplify any further.
So, the sum of the rational expressions 3/(x-1) + 2/(x+2) is (5x + 4)/[(x-1)(x+2)].