Exercise 2 Page 388

Adding and subtracting rational expressions is done using the exact same thought process as adding and subtracting numerical fractions.

Adding or Subtracting With Like Denominators

If two numerical fractions or rational expressions have like denominators, we only need to add or subtract the numerators. Consider the following examples of addition. c|c Numerical & Rational Fractions & Expressions [0.8em] 5/9+2/9 & 2x+1/x-1+- x+5/x-1 Note that in both cases the denominators are the same. Therefore, to simplify the expressions, we need to add the numerators. Let's start with the numerical fractions. Let's now proceed in the same way to add the rational expressions.

2x+1/x-1+- x+5/x-1

AddFrac

Add fractions

2x+1+(- x)+5/x-1

AddNeg

a+(- b)=a-b

2x+1-x+5/x-1

AddSubTerms

Add and subtract terms

x+6/x-1

Adding or Subtracting With Unlike Denominators

There are two ways of adding and subtracting rational expressions and fractions with unlike denominators. Let's explore them one at a time.

First Method

The first method consists of multiplying and dividing each term by the denominator of the other term. This will result in like denominators. Therefore, from here, we can add or subtract the numerators. a/c ± b/d = ad/cd ± bc/cd = ad± bc/cd This method is valid for both numerical fractions and rational expressions. Consider the following examples of subtraction. c|c Numerical & Rational Fractions & Expressions [0.8em] 2/3-1/6 & x+5/x^2-1-x-3/x+1 In both cases, we have unlike denominators. Let's use the method we explained above to perform the subtraction. We will start with the numerical fractions.

2/3-1/6

ExpandFrac

a/b=a * 6/b * 6

2(6)/3(6)-1/6

ExpandFrac

a/b=a * 3/b * 3

2(6)/3(6)-1(3)/6(3)

Multiply

Multiply

12/18-3/18

SubFrac

Subtract fractions

12-3/18

SubTerms

Subtract terms

9/18

ReduceFrac

a/b=.a /9./.b /9.

1/2

Let's now proceed in the same way to subtract the rational expressions.

x+5/x^2-1-x-3/x+1

ExpandFrac

a/b=a * (x+1)/b * (x+1)

(x+5)(x+1)/(x^2-1)(x+1)-x-3/x+1

ExpandFrac

a/b=a * (x^2-1)/b * (x^2-1)

(x+5)(x+1)/(x^2-1)(x+1)-(x^2-1)(x-3)/(x^2-1)(x+1)

SubFrac

Subtract fractions

(x+5)(x+1)-(x^2-1)(x-3)/(x^2-1)(x+1)

▼

Multiply parentheses

Distr

Distribute (x+1)

x(x+1)+5(x+1)-(x^2-1)(x-3)/x^2(x+1)-1(x+1)

Distr

Distribute - 1

x(x+1)+5(x+1)+(- x^2+1)(x-3)/x^2(x+1)-x-1

Distr

Distribute (x-3)

x(x+1)+5(x+1)-x^2(x-3)+1(x-3)/x^2(x+1)-x-1

Distr

Distribute x, 5, - x^2, 1, & x^2

x^2+x+5x+5-x^3+3x^2+x-3/x^3+x^2-x-1

AddSubTerms

Add and subtract terms

- x^3+4x^2+7x+2/x^3+x^2-x-1

In this case, we can factor both the numerator and the denominator. Let's do it and cancel out any possible common factors.

- x^3+4x^2+7x+2/x^3+x^2-x-1

SplitIntoFactors

Split into factors

(x+1)(- x^2+5x+2)/(x+1)(x^2-1)

CancelCommonFac

Cancel out common factors

(x+1)(- x^2+5x+2)/(x+1)(x^2-1)

SimpQuot

Simplify quotient

- x^2+5x+2/x^2-1

Second Method

The second method for adding and subtracting numerical fractions or rational expressions with unlike denominators relies on multiplying all numerators and denominators by the least common denominator. We will consider the same examples as before and subtract using this method. c|c Numerical & Rational Fractions & Expressions [0.8em] 2/3-1/6 & x+5/x^2-1-x-3/x+1 Let's start by subtracting the numerical fractions. Note that 6 is a multiple of 3. This means that the least common denominator is 6.

2/3-1/6

ExpandFrac

a/b=a * 2/b * 2

4/6-1/6

SubFrac

Subtract fractions

4-1/6

SubTerms

Subtract terms

3/6

ReduceFrac

a/b=.a /3./.b /3.

1/2

Notice that we got the same answer as with the first method! Let's now subtract the rational expressions. Let's start by factoring the first denominator.

x+5/x^2-1

WritePow

Write as a power

x+5/x^2- 1^2

FacDiffSquares

a^2-b^2=(a+b)(a-b)

x+5/( x+ 1)( x- 1)

Therefore, the first denominator is a multiple of the second denominator. This means that the least common denominator is (x+1)(x-1). Thus, we need to find an equivalent fraction to x-3x+1 whose denominator is (x+1)(x-1).

x+5/(x+1)(x-1)-x-3/x+1

ExpandFrac

a/b=a * (x-1)/b * (x-1)

x+5/(x+1)(x-1)-(x-3)(x-1)/(x+1)(x-1)

SubFrac

Subtract fractions

x+5-(x-3)(x-1)/(x+1)(x-1)

▼

Multiply parentheses

Distr

Distribute - 1

x+5+(- x+3)(x-1)/(x+1)(x-1)

Distr

Distribute (x-1)

x+5-x(x-1)+3(x-1)/x(x-1)+1(x-1)

Distr

Distribute - x, 3, x, & 1

x+5-x^2+x+3x-3/x^2-x+x-1

AddSubTerms

Add and subtract terms

- x^2+5x+2/x^2-1

Note that, once again, we obtained the same result as before. The advantage of using this method is that we do not need to factor the final expression.