College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

P Prerequisites

P.8 Rational Expressions

Fractional Expression A quotient of two algebraic expressions is called a fractional expression. Here are some examples:

Rational Expression A rational expression is a fractional expression where both the numerator and denominator are polynomials. Here are some examples:

In this section, we learn: Rational Expressions In this section, we learn: How to perform algebraic operations on rational expressions.

The Domain of an Algebraic Expression

The Domain of an Algebraic Expression In general, an algebraic expression may not be defined for all values of the variable. The domain of an algebraic expression is: The set of real numbers that the variable is permitted to have.

The Domain of an Algebraic Expression The table gives some basic expressions and their domains.

E.g. 1—Finding the Domain of an Expression Consider the expression Find the value of the expression for x = 2. Find the domain of the expression.

E.g. 1—The Value of an Expression Example (a) We find the value by substituting 2 for x in the expression:

E.g. 1—Domain of an Expression Example (b) The denominator is zero when x = 3. Since division by zero is not defined: We have x ≠ 3. Thus, the domain is all real numbers except 3. We can write this in set notation as {x | x ≠ 3}

E.g. 2—Finding the Domain of an Expression Find the domains of these expressions.

This polynomial is defined for every x. E.g. 2—Finding the Domain Example (a) 2x2 + 3x – 1 This polynomial is defined for every x. Thus, the domain is the set of real numbers.

We first factor the denominator. E.g. 2—Finding the Domain Example (b) We first factor the denominator. The denominator is zero when x = 2 or x = 3. So, the expression is not defined for these numbers. Hence, the domain is: {x | x ≠ 2 and x ≠ 3}

For the numerator to be defined, we must have x ≥ 0. E.g. 2—Finding the Domain Example (c) For the numerator to be defined, we must have x ≥ 0. Also, we cannot divide by zero; so, x ≠ 5. Thus, the domain is: {x | x ≥ 0 and x ≠ 5}

Simplifying Rational Expressions

Simplifying Rational Expressions To simplify rational expressions, we factor both numerator and denominator and use this property of fractions: This allows us to cancel common factors from the numerator and denominator.

E.g. 3—Simplifying by Cancellation

We can’t cancel the x2’s in Caution We can’t cancel the x2’s in because x2 is not a factor.

Multiplying and Dividing Rational Expressions

Multiplying Rational Expressions To multiply rational expressions, we use this property of fractions: This says that: To multiply two fractions, we multiply their numerators and multiply their denominators.

E.g. 4—Multiplying Rational Expressions Perform the indicated multiplication and simplify:

E.g. 4—Multiplying Rational Expressions We first factor.

Dividing Rational Expressions To divide rational expressions, we use this property of fractions: This says that: To divide a fraction by another fraction, we invert the divisor and multiply.

E.g. 5—Dividing Rational Expressions Perform the indicated division and simplify:

E.g. 5—Dividing Rational Expressions

Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions To add or subtract rational expressions, we first find a common denominator and then use this property of fractions:

Adding and Subtracting Rational Expressions Any common denominator will work. Still, it is best to use the least common denominator (LCD) as learnt in Section P.2. The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

Avoid making the following error: Caution Avoid making the following error:

Caution For instance, if we let A = 2, B = 1, and C = 1, then we see the error:

E.g. 6—Adding and Subtracting Rational Expressions Perform the indicated operations and simplify:

The LCD is simply the product (x – 1)(x + 2). E.g. 6—Adding Example (a) The LCD is simply the product (x – 1)(x + 2).

The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2. E.g. 6—Subtracting Example (b) The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2.

E.g. 6—Subtracting Rational Exp. Example (b)

Compound Fractions

A compound fraction is: A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

E.g. 7—Simplifying a Compound Fraction

One solution is as follows. E.g. 7—Simplifying Solution 1 One solution is as follows. Combine the terms in the numerator into a single fraction. Do the same in the denominator. Invert and multiply.

E.g. 7—Simplifying Solution 1 Thus,

Another solution is as follows. E.g. 7—Simplifying Solution 2 Another solution is as follows. Find the LCD of all the fractions in the expression. Multiply the numerator and denominator by it.

Here, the LCD of all the fractions is xy. E.g. 7—Simplifying Solution 2 Here, the LCD of all the fractions is xy.

Simplifying a Compound Fraction The next two examples show situations in calculus that require the ability to work with fractional expressions.

E.g. 8—Simplifying a Compound Fraction We begin by combining the fractions in the numerator using a common denominator.

E.g. 8—Simplifying a Compound Fraction

E.g. 8—Simplifying a Compound Fraction

E.g. 9—Simplifying a Compound Fraction

Factor (1 + x2)–1/2 from the numerator. E.g. 9—Simplifying Solution 1 Factor (1 + x2)–1/2 from the numerator.

(1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction. E.g. 9—Simplifying Solution 2 (1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction. Therefore, we can clear all fractions by multiplying numerator and denominator by (1 + x2)1/2.

E.g. 9—Simplifying Solution 2 Thus,

Rationalizing the Denominator or the Numerator

Rationalizing the Denominator If a fraction has a denominator of the form we may rationalize the denominator by multiplying numerator and denominator by the conjugate radical

Rationalizing the Denominator This is effective because, by Special Product Formula 1 in Section P.6, the product of the denominator and its conjugate radical does not contain a radical:

E.g. 10—Rationalizing the Denominator Rationalize the denominator: We multiply both the numerator and the denominator by the conjugate radical of , which is .

E.g. 10—Rationalizing the Denominator Thus,

E.g. 11—Rationalizing the Numerator Rationalize the numerator: We multiply numerator and denominator by the conjugate radical .

E.g. 11—Rationalizing the Numerator Thus,

E.g. 10—Rationalizing the Numerator

Avoiding Common Errors

Avoiding Common Errors Don’t make the mistake of applying properties of multiplication to the operation of addition. Many of the common errors in algebra involve doing just that.

Avoiding Common Errors The table states several multiplication properties and illustrates the error in applying them to addition.

Avoiding Common Errors To verify that the equations in the right-hand column are wrong, simply substitute numbers for a and b and calculate each side.

Avoiding Common Errors For example, if we take a = 2 and b = 2 in the fourth error, we have the following result.

Avoiding Common Errors The left-hand side is: The right-hand side is: Since 1 ≠ ¼, the stated equation is wrong.

Avoiding Common Errors You should similarly convince yourself of the error in each of the other equations. See Exercise 113.