Adding Rational Expressions
In order to add or subtract a rational expression, a common denominator must be found first, and then the operation can be carried out in the numerator.
With like denominators, simply add the two numerators to find the sum.
Example 1:
Add the numerators (the denominator does not change).
Add the numerators (the denominator does not change).
With unlike denominators
First find the common denominator
By multiplying denominators
Example 2: Single Term Ratios
Multiply each ratio by one using the other denominator.
Multiply across.
Add the numerators.
Multiply each ratio by one using the other denominator.
Multiply across.
Add the numerators.
By finding least common multiple of the denominators
Example 3: Single Term Ratios
The least common multiple (LCM) from 8x and 5x is 40x
Multiply the ratios by one to get a common denominator.
Multiply across.
Add the numerators, simplify if possible.
163 and 40 are relatively prime, so this ratio cannot be simplified.
The least common multiple (LCM) from 8x and 5x is 40x
Multiply the ratios by one to get a common denominator.
Multiply across.
Add the numerators, simplify if possible.
163 and 40 are relatively prime, so this ratio cannot be simplified.
Example 4: Factoring trinomials in the denominator.
x2 + 5x - 24 = (x + 8)(x - 3)Factor the denominator.
The LCM is (x + 8)(x - 3).
Multiply the second ratio to obtain a common denominator.
Add the numerators.
Use the distributive property to combine like terms.
Rewrite in simplified form.
x2 + 5x - 24 = (x + 8)(x - 3)Factor the denominator.
The LCM is (x + 8)(x - 3).
Multiply the second ratio to obtain a common denominator.
Add the numerators.
Use the distributive property to combine like terms.
Rewrite in simplified form.
Example 5: Special products in the denominator.
x2 - 4 = (x + 2)(x - 2) and x2 + 4x + 4 = (x + 2)(x + 2)Factor the denominators.
The LCM is (x + 2)(x + 2)(x - 2).
Multiply the ratios to obtain common denominators.
Rewrite.
Add the numerators.
Regroup the terms to factor the numerator.
Eliminate common factors.
Rewrite in simplified form.
x2 - 4 = (x + 2)(x - 2) and x2 + 4x + 4 = (x + 2)(x + 2)Factor the denominators.
The LCM is (x + 2)(x + 2)(x - 2).
Multiply the ratios to obtain common denominators.
Rewrite.
Add the numerators.
Regroup the terms to factor the numerator.
Eliminate common factors.
Rewrite in simplified form.
Example 6: Subtraction
x2 - 2x - 8 = (x - 4)(x + 2)Factor the denominators.
The LCM is (x - 8)(x - 4)(x + 2).
Multiply the ratios to obtain common denominators.
Multiply to determine new numerators
Subtract the second numerator from the first.
With -2x - 14 = -2(x + 7) there are no common factors to simplify the ratio; keep the ratio as is.
x2 - 2x - 8 = (x - 4)(x + 2)Factor the denominators.
The LCM is (x - 8)(x - 4)(x + 2).
Multiply the ratios to obtain common denominators.
Multiply to determine new numerators
Subtract the second numerator from the first.
With -2x - 14 = -2(x + 7) there are no common factors to simplify the ratio; keep the ratio as is.
Example 7: Addition and subtraction together
x2 - 25 = (x + 5)(x - 5)Factor the denominator.
The LCM is (x + 5)(x - 5).
Multiply the ratios to obtain a common denominator.
Rewrite the numerator.
Combine like terms.
x2 - 25 = (x + 5)(x - 5)Factor the denominator.
The LCM is (x + 5)(x - 5).
Multiply the ratios to obtain a common denominator.
Rewrite the numerator.
Combine like terms.
Now that you can add and subtract rational expressions, you are ready to start solving rational equations.
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Related Links: Math algebra Adding and Subtracting Rational Expressions Worksheets Rational Expressions Multiplying Rational Equations Simplifying Rational Expressions Algebra Topics |
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