Adding and Subtracting Rational Expressions

Rational Expressions and Equations Playlist on YouTube

When adding or subtracting rational expressions we will need a common denominator.

Simplify the resulting rational expression after adding or subtracting them.

Add or subtract.

If the rational expressions that we are adding or subtracting have unlike denominators then we will need to find the equivalent fractions with the same denominator. To do this multiply both the numerator and denominator of each expression by the factors needed to obtain a common denominator. To help determine the LCD, first factor the denominators.

Instructional Video: Adding and Subtracting Rational Expressions

Perform the operations and state the restrictions to the domain.

For the given functions find f(x) - g(x) and state the restrictions to the domain.

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Multiplying and Dividing Rational Expressions

Multiplying rational expressions is similar to simplifying them.  We will factor then cancel.

Rational Expressions and Equations Playlist on YouTube

Only cancel factors that are the same, or opposites, in the numerator and denominator.  Recall the property for multiplying fractions,

When multiplying fractions there is no need for a common denominator, just multiply the numerators and denominators and then simplify.

Multiply.

The previous questions did not ask for the restrictions but we certainly can list them anyway. Look at the factors in the denominator to see what values for x will evaluate it to zero. Remember that the function notation that implies multiplication.

For the given functions find (f *g)(x).

When dividing fractions there is no need for a common denominator.

However, the above property reminds us that when dividing by a fraction the result will be the same as multiplying by the reciprocal of that fraction. At this point, we will add a step when dividing; we need to reciprocate, factor then cancel.

Divide.

The list of restrictions in the previous problem is a bit more involved. As before, look at all the factors in the denominator, even if it was cancelled, to find the values that evaluate to zero.

Look at the denominators in each step to identify the restrictions.

Remember that the function notation that implies division.

For the given functions find (f / g)(x).

Video Examples on YouTube:

Chapter 6 Sample Test Questions

Click here for a worksheet containing 20 sample test questions with answers.

[ Sample Test Questions Chapter 6 ]

Example: Given the function:

   a. Calculate f(0).

   b. Find all values of x for which f(x) = 0.

 Example: Find a quadratic equation with integer coefficients and the following solution set.

Example: The product of two consecutive odd positive integers is eleven less than 10 times the larger. Find the integers.

Example: The length of a rectangle is 6 centimeters longer than twice its width. If the area is 140 square centimeters, find the dimensions of the rectangle.

Example: The cost of a particular car in dollars can be approximated by the function:

Here x represents the age of the car in years.

   a. Use the graph to determine the cost of the car when it was new?

   b. How old will the car be when it reaches its minimum cost?

   c. How much is this car worth when it reaches 5 years old?