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Chapter 8 Rational Expressions

8.1 Simplifying Rational Expressions 8.2 Multiplying and Dividing Rational Expressions 8.3 Finding the Least Common Denominator 8.4Adding and Subtracting Rational Expressions Putting It All Together 8.5Simplifying Complex Fractions 8.6Solving Rational Equations 8.7Applications of Rational Equations 8 Rational Expressions

Simplifying Rational Expressions8.1 Evaluate a Rational Expression In Section 1.4, we defined a rational number as the quotient of two integers where the denominator does not equal zero. Some examples of rational numbers are We can define a rational expression in a similar way. A rational expression is a quotient of two polynomials provided that the denominator does not equal zero. Some examples of rational expressions are We can evaluate rational expressions for given values of the variables as long as the values do not make any denominators equal zero.

Example 1 Solution Substitute 4 for x. Substitute 5 for x. Substitute 1 for x. Undefined A fraction is undefined when its denominator equals zero. Therefore, we say is undefined when x = 1 since this value of x makes the denominator equal zero. So, x cannot equal 1 in this expression.

Finding the Values of the Variable That Make a Rational Expression Undefined or Equal to Zero Two important aspects of fractions and rational expressions are listed below. Example 1-Continued For each rational expression, for what values of the variable i)Does the expression equal zero? ii)Is the expression undefined? Solution i)The rational expression equals zero when the numerator equals zero. Set the numerator equal to zero, and solve for y. Part ii continued on next slide…

Example 2 For each rational expression, for what values of the variable i)Does the expression equal zero? ii)Is the expression undefined? Solution ii)The rational expression is undefined when its denominator equals zero. Set the denominator equal to zero, and solve for y. This means that any real number except -9 can be substituted for y in this expression.

Example 3 For each rational expression, for what values of the variable i)Does the expression equal zero? ii)Is the expression undefined? Solution i)The rational expression equals zero when the numerator equals zero. Set the numerator equal to zero, and solve for r. ii)The rational expression is undefined when its denominator equals zero. Set the denominator equal to zero, and solve for r. This means that any real number except -8 and 3 can be substituted for r in this expression. Factor. Set each factor equal to zero. Solve. or

Example 4 For each rational expression, for what values of the variable i)Does the expression equal zero? ii)Is the expression undefined? Solution i)The rational expression equals zero when the numerator equals zero. The numerator is 7, and. ii)The rational expression is undefined when its denominator equals zero. Set the denominator equal to zero, and solve for p.

Example 5 For each rational expression, for what values of the variable i)Does the expression equal zero? ii)Is the expression undefined? Solution i)The rational expression equals zero when the numerator equals zero. Set numerator equal to zero and solve for d. ii)The rational expression is undefined when its denominator equals zero. However, the denominator is 4 and. Factor. Set each factor equal to zero. Solve.

Write a Rational Expression in Lowest Terms The approach for writing a fraction in lowest terms can be used to write a Rational Expression in lowest terms. Let’s refresh the approach used when writing a fraction In lowest terms. This approach will be used to write a rational expression in lowest terms.

This property mirrors the example on the previous slide since Or, we can also think of the reducing procedure as dividing the numerator and denominator by the common factor C.

Example 6 Solution Write each rational expression in lowest terms. Divide 21 by 3 and use the quotient rule. Factor numerator and denominator. Divide out the common factor (m-3). Factor. Factor completely. Divide out the common factor (z+4).

Simplify a Rational Expression of the Form Factor -1 out of denominator. Rewrite (-3 + x) as (x – 3). Divide out the common factor, x – 3.

Example 7 Write each rational expression in lowest terms. Solution Factor. Distribute Factor numerator and denominator. The negative sign can be written in front of the fraction.

Write Equivalent Forms of a Rational Expression The answer to Example 7c can be written in different ways. The negative sign in front of a fraction can be applied to the numerator or to the denominator. Below are the different ways that this rational expression can be written in equivalent forms. Apply negative sign to denominator. Apply negative sign to numerator. Keep this idea of equivalent form of rational expressions as you work through the practice problems.

Determine the Domain of a Rational Function We can combine what we have learned about rational expressions with what we Learned about functions in Section 4.6. is an example of a rational function since is a rational expression and since each value that can be substituted for x will produce only one value for the expression. Recall from Chapter 4 that the domain of a function f(x) is the set of all real numbers That can be substituted for x. Since a rational expression is undefined when its Denominator equals zero, we define the domain of a rational function as follows. Therefore, to determine the domain of a rational function we set the denominator equal to zero and solve for the variable. Any value that makes the denominator equal to zero is not in the domain of the function.

Example 7 Determine the domain of each rational function. Solution a)f(x) is undefined when the denominator equals zero. Set the denominator equal to zero, and solve for x. Set the denominator = 0. Solve. When x = -5, the denominator equals zero. The domain contains all real numbers except -5. Write the domain in interval notation as b) f(k) is undefined when the denominator equals zero. Set the denominator equal to zero, and solve for k. Set the denominator = 0. Factor. Set each factor equal to 0. or When k = 6 or k = 2, the denominator equals zero. The domain contains all real numbers except 6 and 2. Write the domain in interval notation as

Example 7 Determine the domain of each rational function. Solution a)h(c) is undefined when the denominator equals zero. Set the denominator equal to zero, and solve for c. Notice that in the denominator we have the number 4. The number 4 will never Equal zero. Therefore, there is no value of c that will make this rational function Undefined. Any real number may be substituted for c and the function will be Defined. The domain of the function is the set of all real numbers. We can write down in Interval notation as