Topic 3: Adding and Subtracting Rational Expressions

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.

This explore activity is different than the one found in your book This explore activity is different than the one found in your book. Please complete this on a separate sheet of paper. Explore… 1. Add 1 2 and 2 3 . 2. Explain in detail the strategy you used for adding these rational numbers. 3. If you were asked to subtract the same two rational numbers, how would the strategy change? 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!

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

Information… Lets’ look at an example: Factors of 12a2: 2,2,3,a,a This is called prime factorization. It is the breakdown of the numbers into prime factors and the listing of all variable factors. Factors of 12a2: 2,2,3,a,a Factors of 15a(a-2): 3,5,a,(a-2) 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

Information… Lets’ look at an example: Factors of 12a2: 2,2,3,a,a Once you’ve listed the factors, make a list of the ones you need to cover all factors in all denominators. Lets’ look at an example: We need 2,2,3,a and a to cover the 12a2. We need the additional factor of (a-2) to cover the 6a(a-2). We don’t need additional factors of 2, 3, or a since they are already covered. Factors of 12a2: 2,2,3,a,a Factors of 6a(a-2): 2,3,a,(a-2) The LCD needs to contain the factors 2, 2, 3, a, a, and a-2. The LCD, then, is 12a2(a-2)

Information… Lets’ try another… Factors of 14xy: 2,7,x,y We need 2,7,x,and y to cover the 14xy. We need the additional factors of 3, x, and z to cover the 21x2yz. We don’t need additional factors of 7, x, or y since they are already covered. Factors of 14xy: 2,7,x,y Factors of 21x2yz: 3,7,x,x,y,z The LCD needs to contain the factors 2, 7, x, y, 3, x, and z. The LCD, then, is 42x2yz.

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.

Example 1 Simplifying sums and differences with monomial denominators Find the LCD and the NPVs for each question. Then simplify. a) Factors of 2: 2 3x 4x Factors of 3: 3 The LCD needs to contain the factors 2 and 3. The LCD, then, is 6.

Example 1 b) 10x2 12 Factors of 3: 3 Factors of 5x: 5,x Simplifying sums and differences with monomial denominators b) 10x2 12 Factors of 3: 3 Factors of 5x: 5,x The LCD needs to contain the factors 3, 5, and x. The LCD, then, is 15x. x≠0 Remember: The LCD must contain the greatest number of any factor that appears in the denominator of either fraction. There are no factors common to the numerator and denominator.

Example 1 Simplifying sums and differences with monomial denominators c) The NPV 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.

Example 1 Simplify the following sums and differences d) Simplifying sums and differences with monomial denominators Simplify the following sums and differences d) The NPV Factors of 8x2: 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 8x2.

Example 2 Simplify the following sums and differences. a) Simplifying sums and differences with binomial denominators Simplify the following sums and differences. a) The NPV 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).

Example 2 Simplify the following sums and differences. b) Simplifying sums and differences with binomial denominators Simplify the following sums and differences. b) The NPV You don’t need to come up with an LCD, since they already have the same denominator.

Example 2 Simplify the following sums and differences. c) Simplifying sums and differences with binomial denominators Simplify the following sums and differences. c) The NPV 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).

Example 3 Simplify the following expressions. a) The NPV Careful! The first fraction’s denominator can be factored as the difference of squares (x-4)(x+4) Using a factoring strategy to simplify a rational expression Simplify the following expressions. a) The NPV Factors of x2-16: (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).

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

Example 3 Simplify the following expressions. b) The NPV Careful! The first fraction’s denominator can be factored as the difference of squares (x-1)(x+1) Simplify the following expressions. b) The NPV Factors of x2-1: (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).

Example 3 Simplify the following expressions. c) The NPV Careful! The first fraction’s denominator can be factored as the difference of squares (x-1)(x+1) Simplify the following expressions. c) The NPV Factors of 4a+2: 2(2a+1) Factors of 4a2-1: (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).

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.

Adding and Subtracting Rational Expressions Need to Know You’re ready! Try the homework from this section. 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.