Learning Resources
Lesson
We multiply rational expressions in much the same way that we multiply rational numbers or fractions. The multiplication rule for fractions is:
This same rule applies regardless whether p, q, r, and s represent numbers or polynomials.
Since the product of rational expressions should be written in simplest form (reduced to lowest terms), we can shorten the multiplication process by dividing the numerator and denominator by any common factors of both before we multiply the numerators together and the denominators together. The process is shown in the following examples.
Example 1
Multiply and express the answer in simplest terms: 
Solution
Example 2
Multiply and express the answer in simplest terms: 
Solution
Example 3
Multiply and express the answer in simplest terms: 
Solution
We divide rational expressions in much the same way that we divide rational numbers or fractions. The division rule for fractions is:
This means that to divide we multiply by the reciprocal of the divisor. This same rule applies regardless whether p, q, r, and s represent numbers or polynomials.
To divide rational expressions we simply rewrite it as a multiplication problem and proceed as for the multiplication problems shown on page a of this lesson.
Example 1
Divide and express the answer in simplest terms: 
Solution
Take particular note of the numbers that must be excluded from the replacement set when dividing. Obviously, the denominators of the original fractions can not be zero. Thus, in Example 1 above, the denominator of the first fraction can not be zero, i.e. y2 - 9 ¹ 0, and the denominator of the second fraction (the divisor) cannot be zero, i.e. y + 3 ¹ 0 .
However, also note that the numerator of the divisor can not be zero. In Example 1 above this means 2y - 10 ¹ 0 . This is because we have to multiply by its reciprocal, thus making 2y - 10 a denominator in the multiplication and thus it cannot be zero.
Example 2
Divide and express the answer in simplest terms: 
Solution
Activity
For each of the following, perform the indicated operations and express the answer in simplest terms.
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Test Yourself
For each of the following, perform the indicated operations and express the answer in simplest terms. Don't forget the replacement set.
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