Showing posts with label rational. Show all posts
Showing posts with label rational. Show all posts

Tuesday, November 6, 2012

Adding and Subtracting Rational Expressions

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.
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.  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).
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Simplifying Rational Expressions


Given a rational expression, the quotient of two polynomials, we will factor the numerator and denominator if we can and then cancel factors that are exactly the same.
When evaluating rational expressions, plug in the appropriate values either before simplifying or after, the result will be the same.  Although, it is more efficient to simplify first then evaluate.
We can see that when evaluating, the result will be the same whether or not we simplify first.  It turns out that not all numbers can be used when we evaluate.
The point is that not all real numbers will be defined in the above rational expression.  In fact, there are two restrictions to the domain, -2 and 3/5.  These values, when plugged in, will result in zero in the denominator.  Another way to say this is that the domain consists of all real numbers except for −2 and 3/5.

Tip: To find the restrictions, set each factor in the denominator equal to zero and solve. The factors in the numerator do not contribute to the list of restrictions.

Simplify and state the restrictions to the domain.


Even if the factor cancels it still contributes to the list of restrictions.  Basically, it is important to remember the domain of the original expression when simplifying. Also, we must use caution when simplifying, please do not try to take obviously incorrect shortcuts like this:

Since subtraction is not commutative, we must be alert to opposite binomial factors.  For example, 5 − 3 = 2 and 3 − 5 = −2. In general,
Simplify and state the restrictions to the domain.
At this point, we evaluate using function notation.
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