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**Adding and Subtracting Rational Expressions**

Section 7.4 beginning on page 384

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**Adding/Subtracting With Like Denominators**

When the denominators are the same, add or subtract the numerator and keep the denominator. Example 1: a) 7 4π₯ + 3 4π₯ b) 2π₯ π₯+6 β 5 π₯+6 = 7+3 4π₯ = 10 4π₯ = 5 2π₯ = 2π₯β5 π₯+6

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**Finding the Least Common Multiple**

Example 2: Find the least common multiple of 4 π₯ 2 β16 and 6 π₯ 2 β24π₯+24 ** Factor each polynomial and write numerical factors as products of primes. 4 π₯ 2 β16 6 π₯ 2 β24π₯+24 =4( π₯ 2 β4) =6( π₯ 2 β4π₯+4) =2β3(π₯β2)(π₯β2) = 2 2 (π₯+2)(π₯β2) =2β3 (π₯β2) 2 **The LCM is the product of the highest power of each factor that appears in either polynomial. πΏπΆπ= β3(π₯+2) (π₯β2) 2 =12(π₯+2) (π₯β2) 2

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**Adding with Unlike Denominators**

Example 3: Find the sum 7 9 π₯ 2 + π₯ 3 π₯ 2 +3π₯ Step 1: Create equivalent fractions that have the same denominator. You can always find a common denominator by multiplying the denominators together. If you use the LCM simplifying at the end may be easier but the creating equivalent fractions will be harder. 3 π₯ 2 +3π₯ 3 π₯ 2 +3π₯ β 7 9 π₯ 2 + π₯ 3 π₯ 2 +3π₯ β 9 π₯ 2 9 π₯ 2 = 7(3 π₯ 2 +3π₯) 9 π₯ 2 (3 π₯ 2 +3π₯) + 9 π₯ 2 (π₯) 9π₯ 2 (3 π₯ 2 +3π₯) Step 2: Add the numerators, simplify the numerator. Step 3: Factor the numerator and denominator. Step 4: Simplify. = 7 3 π₯ 2 +3π₯ +9 π₯ 2 (π₯) 9 π₯ 2 (3 π₯ 2 +3π₯) = 9 π₯ π₯ 2 +21π₯ 9 π₯ 2 (3 π₯ 2 +3π₯) = 3π₯(3 π₯ 2 +7π₯+7) 9 π₯ 2 β3π₯(π₯+1) = 3 π₯ 2 +7π₯+7 9 π₯ 2 (π₯+1)

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**Subtracting With Unlike Denominators**

Example 4: Find the difference π₯+2 2π₯β2 β β2π₯β1 π₯ 2 β4π₯β3 It can be helpful to factor the denominators first to allow you to multiply the fractions by fewer factors. (by only multiplying by the factors that are not already in both denominators) = π₯+2 2(π₯β1) β β2π₯β1 (π₯β3)(π₯β1) (π₯β3) (π₯β3) β β 2 2 = π₯ 2 βπ₯β6 2(π₯β3)(π₯β1) β β4π₯β2 2(π₯β3)(π₯β1) = π₯ 2 βπ₯β6β(β4π₯β2) 2(π₯β3)(π₯β1) = π₯ 2 βπ₯β6+4π₯+2 2(π₯β3)(π₯β1) = π₯ 2 +3π₯β4 2(π₯β3)(π₯β1) = (π₯β1)(π₯+4) 2(π₯β3)(π₯β1) = π₯+4 2(π₯β3) ;π₯β 1

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**Simplifying Complex Fractions**

5 π₯ π₯ π₯ Example 6: Simplify 5 π₯ π₯ π₯ Step 1: Simplify the numerator and denominator so that there is only one fraction in each. Step 2: To divide, multiply the numerator by the reciprocal of the denominator. (π₯+4) (π₯+4) π₯ π₯ β = 5 π₯+4 π₯ π₯(π₯+4) + 2π₯+8 π₯(π₯+4) = 5 π₯+4 3π₯+8 π₯(π₯+4) = 5 π₯+4 β π₯(π₯+4) 3π₯+8 = 5π₯ 3π₯+8 ;π₯β β4, 0

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**Monitoring Progress Find the sum or difference:**

1) 8 12π₯ β 5 12π₯ 2) 2 3 π₯ π₯ 2 3) 4π₯ π₯β2 β π₯ π₯β2 4) 2 π₯ 2 π₯ π₯ 2 +1 Find the least common multiple: 5) 5 π₯ 3 ,10 π₯ 2 β15π₯ 6) 3 4π₯ β ) π₯ 2 + π₯ 9π₯ 2 β12 8) π₯ π₯ 2 βπ₯β π₯β48 = 1 4π₯ = 1 π₯ 2 = 3π₯ π₯β2 =2 πΏπΆπ=5 π₯ 3 (2π₯β3) = 21β4π₯ 28π₯ = (π₯β1) (π₯+2) 2 3 π₯ 2 (3 π₯ 2 β4) = 17π₯+15 12(π₯β4) (π₯+3)

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**Monitoring Progress Simplify the complex fractions. 10)**

11) 12) π₯ 6 β π₯ 3 π₯ 5 β 7 10 = β5π₯ 3(2π₯β7) 2 π₯ β4 2 π₯ +3 = 2(1β2π₯) 2+3π₯ 3 π₯ π₯β3 + 1 π₯+5 = 3(π₯β3) 3π₯+7

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