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Essential Question: How do you add and subtract two rational expressions?

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Just like with fractions, terms need to have a common denominator in order for their numerators to be added or subtracted together. After they’re added/subtracted, fractions can be simplified. Sometimes, you will need to find the least common denominator (LCD). To do this, you find the least common multiple of the denominators.

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To find the least common multiple 1) Find the prime factors of each expression 2) Write each prime factor the greatest number of times it appears in either expression 3) Simplify where possible Find the LCM of 4x 2 – 36 and 6x x + 54 Step 1 4x 2 – 36 6x x + 54 Step 2 Step 3 = 4(x 2 – 9)= (2)(2) = 6(x 2 + 6x + 9)= (2) (2) appears twice, (3) appears once, (x – 3) appears once,(x + 3) appears twice (x – 3) (x + 3) (3) (x + 3)(x + 3) (2)(2)(3)(x – 3)(x + 3)(x + 3) = 12(x – 3)(x + 3) 2

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Your Turn #1 Find the LCM of 3x 2 – 9x - 30 and 6x + 30 Step 1 3x 2 – 9x – 30 6x + 30 Step 2 Step 3 = 3(x 2 – 3x – 10) = (3) = 6(x + 5)= (2) (2) appears once, (3) appears once, (x - 5) appears once, (x + 2) appears once (x – 5) (x + 2) (3)(x + 5) (2)(3)(x – 5)(x + 5)(x + 2) = 6(x – 5)(x + 5)(x + 2) (x + 5) appears once,

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Your Turn #2 Find the LCM of 5x x + 10 and 2x 2 – 8 Step 1 5x x x 2 – 8 Step 2 Step 3 = 5(x 2 + 3x + 2) = (5) = 2(x 2 – 4)= (2) (2) appears once, (5) appears once, (x + 2) appears once, (x – 2) appears once (x + 2) (x + 1) (x + 2) (x – 2) (2)(5)(x + 2)(x + 1)(x – 2) = 10(x + 2)(x + 1)(x – 2) (x + 1) appears once,

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Now that we understand how to find the least common multiple (which is also our least common denominator), let’s use that to add/subtract fractions We multiply each term in the problem by what is missing from the LCD

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Simplify Find the LCM (x + 4)(x + 1) with (3)(x + 1) = (3)(x + 4)(x + 1) Multiply left side (top and bottom) by (3) Multiply right side (top and bottom) by (x + 4)

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Simplify Find the LCM Multiply left side (top and bottom) by Multiply right side (top and bottom) by (x – 6)(x + 2) with (2)(2)(x + 2) = (4)(x – 6)(x + 2) (4) (x - 6)

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Simplify Find the LCM Multiply left side (top and bottom) by Multiply right side (top and bottom) by 5(y + 5)(y – 5) with (3)(y + 5) = (3)(5)(y + 5)(y – 5) (3) 5(y - 5)

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Page 517 – 518 Problems 5 – 21 (odd) Show your work!!!

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Essential Question: How do you add and subtract two rational expressions?

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A complex fraction is a fraction that has a fraction in its numerator, denominator, or both. A few examples: To simplify a complex fraction, multiply all terms by the LCD of all embedded fractions

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Simplify This function has two embedded denominators, x and y, so the LCD of all embedded denominators is xy. Multiply all terms by xy.

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Simplify Multiply all terms by: x

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Simplify Multiply all terms by: 2y

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Simplify Multiply all terms by: x(x + 1)(x – 1)

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Page 517 – 518 Problems 22 – 30 (all) Show your work!!!

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