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Rational Expressions Topic 3: Adding and Subtracting Rational Expressions

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I can compare the strategies for performing a given operation on rational expressions to the strategies for performing the same operation on rational expressions. I can determine the non-permissible values when performing operations on rational expressions. I can determine, in simplified form, the sum or difference of rational expressions that have the same denominator. I can determine, in simplified form, the sum or difference of two rational expressions that have different denominators.

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Explore You need to determine the lowest common denominator, and then adjust each fraction to have the LCD. Then you add the numerators and keep the denominator the same. You would subtract the numerators instead of adding them! This explore activity is different than the one found in your book. Please complete this on a separate sheet of paper.

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Information In the past you have found Lowest Common Denominator by listing the multiples of each denominator and then identifying the first one that is common to both. This is problematic in Rational Expressions. For this unit, we will handle lowest common denominators in a different way. We will instead: Identify all factors of each denominator Identify a common denominator that contains all factors

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Information Lets look at an example: Factors of 12a 2 : 2,2,3,a,a Factors of 15a(a-2): 3,5,a,(a-2) This is called prime factorization. It is the breakdown of the numbers into prime factors and the listing of all variable factors. For help with breaking the numbers into their prime factors, try the following site: http://nlvm.usu.edu/en/nav/frames_asid_202_g_3_t_1.html

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Information Lets look at an example: Factors of 12a 2 : 2,2,3,a,a Factors of 6a(a-2): 2,3,a,(a-2) Once youve listed the factors, make a list of the ones you need to cover all factors in all denominators. We need 2,2,3,a and a to cover the 12a 2. We need the additional factor of (a-2) to cover the 6a(a-2). We dont need additional factors of 2, 3, or a since they are already covered. The LCD needs to contain the factors 2, 2, 3, a, a, and a-2. The LCD, then, is 12a 2 (a-2)

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Information Lets try another Factors of 14xy: 2,7,x,y Factors of 21x 2 yz: 3,7,x,x,y,z We need 2,7,x,and y to cover the 14xy. We need the additional factors of 3, x, and z to cover the 21x 2 yz. We dont need additional factors of 7, x, or y since they are already covered. The LCD needs to contain the factors 2, 7, x, y, 3, x, and z. The LCD, then, is 42x 2 yz.

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Information The strategies used to add and subtract rational numbers can be used to add and subtract rational expressions. Find the lowest common denominator for all of the rational expressions. Find equivalent expressions for all of the rational expressions so that everything has the same denominator. Add or subtract the numerators, and keep the denominator the same. Simplify, stating all restrictions Any polynomial that ever appears in the denominator of a rational expression is used to determine the non-permissible values of the rational expression.

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Example 1 Find the LCD and the NPVs for each question. Then simplify. a) Simplifying sums and differences with monomial denominators Factors of 2: 2 Factors of 3: 3 The LCD needs to contain the factors 2 and 3. The LCD, then, is 6. 3x4x

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b) Example 1 Simplifying sums and differences with monomial denominators Remember: The LCD must contain the greatest number of any factor that appears in the denominator of either fraction. Factors of 5x: 5,x Factors of 3: 3 The LCD needs to contain the factors 3, 5, and x. The LCD, then, is 15x. x 0 10x 2 12 There are no factors common to the numerator and denominator.

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Example 1 c) Simplifying sums and differences with monomial denominators Factors of 3b: 3,b Factors of 5b: 5, b The LCD needs to contain the factors 3, b, and 5. The LCD, then, is 15b. The NPV

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Example 1 Simplify the following sums and differences d) Simplifying sums and differences with monomial denominators Factors of 8x 2 : 2,2,2,x,x Factors of 4x: 2,2,x The LCD needs to contain the factors 2, 2, 2, x, and x. The LCD, then, is 8x 2. The NPV

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Example 2 Simplify the following sums and differences. a) Simplifying sums and differences with binomial denominators Factors of x: x Factors of x-4: x-4 The LCD needs to contain the factors x and (x-4). The LCD, then, is x(x-4). The NPV

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Example 2 Simplify the following sums and differences. b) Simplifying sums and differences with binomial denominators You dont need to come up with an LCD, since they already have the same denominator. The NPV

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Example 2 Simplify the following sums and differences. c) Simplifying sums and differences with binomial denominators Factors of n-3: (n-3) Factors of n+2: (n+2) The LCD needs to contain the factors (n-3) and (n+2). The LCD, then, is (n-3)(n+2). The NPV

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Example 3 Simplify the following expressions. a) Using a factoring strategy to simplify a rational expression Factors of x 2 -16: (x-4)(x+4) Factors of x+4: (x+4) The LCD needs to contain the factors (x-4) and (x+4). The LCD, then, is (x-4)(x+4). The NPV Careful! The first fractions denominator can be factored as the difference of squares (x-4)(x+4)

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Example 3 Continued. a) Using a factoring strategy to simplify a rational expression Careful! This one can be simplified further!

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Example 3 Simplify the following expressions. b) Factors of x 2 -1: (x-1)(x+1) Factors of x-1: (x-1) The LCD needs to contain the factors (x-1) and (x+1). The LCD, then, is (x-1)(x+1). The NPV Careful! The first fractions denominator can be factored as the difference of squares (x-1)(x+1)

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Example 3 Simplify the following expressions. c) Factors of 4a+2: 2(2a+1) Factors of 4a 2 -1: (2a+1)(2a-1) The LCD needs to contain the factors 2, (2a-1) and (2a+1). The LCD, then, is 2(2a-1)(2a+1). The NPV Careful! The first fractions denominator can be factored as the difference of squares (x-1)(x+1)

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Need to Know The strategies used to add and subtract rational numbers can be used to add and subtract rational expressions. When rational expressions are added or subtracted, they must have a common denominator.

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Need to Know Steps to add and subtract rational expressions: Adding and Subtracting Rational Expressions 1. Factor all numerators, if possible. 2. Factor all denominators, if possible. 3. Identify the NPVs of the variable in any expression that is at any time in the denominator. 4. Determine the LCD. 5. Rewrite each rational expression as an equivalent expression with the LCD as the denominator. 6. Add or subtract the numerators, and rewrite the denominator. 7. Collect like terms, and then factor the numerator if possible. 8. Simplify using common factors. Remember to state the restrictions. Youre ready! Try the homework from this section.