Rational Expressions Topic 3: Adding and Subtracting Rational
Expressions
Slide 2
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.
Slide 3
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.
Slide 4
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
Slide 5
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
Slide 6
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)
Slide 7
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.
Slide 8
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.
Slide 9
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
Slide 10
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.
Slide 11
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
Slide 12
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
Slide 13
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
Slide 14
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
Slide 15
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
Slide 16
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)
Slide 17
Example 3 Continued. a) Using a factoring strategy to simplify
a rational expression Careful! This one can be simplified
further!
Slide 18
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)
Slide 19
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)
Slide 20
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.
Slide 21
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.