1. Adding and Subtracting Rational Expressions

2. Rational Functions
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
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3. Add rational expressions having the same denominator.
Objective 1
Add rational expressions having
the same denominator.
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4. Add rational expressions having the same denominator.
If and (Q ≠ 0) are rational expressions, then
That is, to add rational expressions with the same denominator, add the numerators and keep the same denominator.
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5. EXAMPLE 1 Add. Write each answer in lowest terms. Solution:
Adding Rational Expressions with the Same Denominator
Add. Write each answer in lowest terms.
Solution:
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6. Add rational expressions having different denominators.
Objective 2
Add rational expressions having
different denominators.
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7. Add rational expressions having different denominators.
Step 1: Find the least common denominator (LCD).
Step 2: Rewrite each rational expression as an equivalent rational expression with the LCD as the denominator.
Step 3: Add the numerators to get the numerator of the sum The LCD is the denominator of the sum.
Step 4: Write in lowest terms using the fundamental property.
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8. EXAMPLE 2 Add. Write each answer in lowest terms. Solution:
Adding Rational Expressions with Different Denominators
Add. Write each answer in lowest terms.
Solution:
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9. EXAMPLE 3 Add. Write the answer in lowest terms. Solution:
Adding Rational Expressions
EXAMPLE 3
Add. Write the answer in lowest terms.
Solution:
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10. EXAMPLE 4 Add. Write the answer in lowest terms. Solution:
Adding Rational Expressions
EXAMPLE 4
Add. Write the answer in lowest terms.
Solution:
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11. EXAMPLE 5 Add. Write the answer in lowest terms. Solution:
Adding Rational Expressions with Denominators That Are Opposites
EXAMPLE 5
Add. Write the answer in lowest terms.
Solution:
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12. Subtract rational expressions.
Objective 3
Subtract rational expressions.
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13. Subtract rational expressions.
Use the following rule to subtract rational expressions having the same denominator.
If and (Q ≠ 0) are rational expressions, then
That is, to subtract rational expressions with the same denominator, subtract the numerators and keep the same denominator.
We subtract rational expressions having different denominators using a procedure similar to the one used to add rational expressions having different denominators.
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14. EXAMPLE 6 Subtract. Write the answer in lowest terms. Solution:
Subtracting Rational Expressions with the Same Denominator
Subtract. Write the answer in lowest terms.
Solution:
Sign errors often occur in subtraction problems. The numerator of the fraction being subtracted must be treated as a single quantity. Be sure to use parentheses after the subtraction sign.
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15. EXAMPLE 7 Subtract. Write the answer in lowest terms. Solution:
Subtracting Rational Expressions with Different Denominators
Subtract. Write the answer in lowest terms.
Solution:
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16. EXAMPLE 8 Subtract. Write the answer in lowest terms. Solution:
Subtracting Rational Expressions with Denominators That Are Opposite
Subtract. Write the answer in lowest terms.
Solution:
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17. EXAMPLE 9 Subtract. Write the answer in lowest terms. Solution:
Subtracting Rational Expressions
Subtract. Write the answer in lowest terms.
Solution:
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