11:59 PM Mountain Time on Monday evening
Intellectual Elaboration:
Hello Class: Welcome to Week 1!
Let’s start by talking about rational expressions.
A rational expression is any expression that can be written as a quotient of two
polynomials. Here are some examples:
Rational expressions are examples of algebraic fractions. They are also examples of
fractional expressions. Because rational expressions indicate division, we must be
careful to avoid denominators that are 0.
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Let’s consider this example
Therefore the expression is undefined at x= 7 and x = -4
Example 1 (Bittinger & Ellenbogen, 2006)
A rational expression is said to be
simplified
when the numerator and the denominator
have no factors (other than 1) in common.
To simplify a rational expression, we first factor the numerator and denominator. We
then identify factors common to the numerator and denominator, rewrite the expression
as a product of two rational expressions (one of which is equal to 1), and then remove
the factor equal to 1.
Canceling is a shortcut that can be used—and easily
misused
—to simplify rational
expressions. Canceling must be done with care and understanding. Essentially,
canceling streamlines the process of removing a factor equal to 1.
Working with rational expressions is exactly the same as working with fractions. All the
same rules are in play.
The Product of Two Rational Expressions
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To multiply rational expressions, multiply numerators and multiply denominators:
The Quotient of Two Rational Expressions
To divide by a rational expression, multiply by its reciprocal:
The Sum of Two Rational Expressions:
To add when the denominators are the same,
add the numerators and keep the common denominator:
The Difference of Two Rational Expressions: To subtract when the denominators are
the same, subtract the second numerator from the first and keep the common
denominator:
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Least Common Multiples and Denominators:
To add or subtract rational expressions that have different denominators, we must first
find equivalent rational expressions that
do
have a common denominator.
The least common multiple must include the factors of each number, so it must include
each prime factor the greatest number of times that it appears in any factorizations.
To Find the Least Common Denominator (LCD)
1.
Write the prime factorization of each denominator.
2.
Select one of the factorizations and inspect it to see if it contains the other.