1. Rational Expressions and Functions: Adding and Subtracting
6.2
Rational Expressions and Functions: Adding and Subtracting
When Denominators are the Same
When Denominators are Different
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2. When Denominators Are the Same
Addition and Subtraction with Like Denominators
To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator.
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3. Example Solution Add: Adding the numerators and combining like terms
Factoring and looking for common factors
Removing a factor equal to 1
Simplifying
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4. Recall that a fraction bar is a grouping symbol
Recall that a fraction bar is a grouping symbol. The next example shows that when a numerator is subtracted, care must be taken to subtract, or change the sign of, each term in that polynomial.
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5. Example
Perform the indicated operation. Simplify, if possible.
Solution
Parentheses remind us to subtract both terms.
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6. When Denominators Are Different
In order to add rational expressions such as
we must first find common denominators. As in arithmetic, our work is easier when we use the least common multiple (LCM) of the denominators.
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7. Least Common Multiple
To find the least common multiple (LCM) of two or more expressions, find the prime factorization of each expression and form a product that contains each factor the greatest number of times that it occurs in any one prime factorization.
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8. Example Solution Find the least common multiple.
We factor both expressions:
By multiplying the factors of x2 + 5x + 6 by x – 7, we form a product that contains both x2 +5x +6 and x2 – 4x – 21 as factors:
LCM = (x + 2)(x + 3)(x – 7)
x2 + 5x + 6 is a factor.
There is no need to multiply this out.
x2 – 4x – 21 is a factor.
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9. To Add or Subtract Rational Expressions
Determine the least common denominator (LCD) by finding the least common multiple of the denominators.
Rewrite each of the original rational expressions, as needed, in an equivalent form that has the LCD.
Add or subtract the resulting rational expressions, as indicated.
Simplify the result, if possible, and list any restrictions on the domain of functions.
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10. Example Solution Add: The LCD is (x + 3)(x + 2)(x – 7).
Multiplying by 1 to get the LCD in each expression.
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11. Solution (continued) We leave the denominator in factored form.
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12. Example Solution Subtract:
Noting that 3y2 – 1 and 1 – 3y2 are opposites.
Performing the multiplication.
Subtracting. The parentheses are important.
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