A rational expression (algebraic fraction) has the same form as dividing two polynomials.
For example: |\dfrac{2x+3}{x^{2}+6x+8}|
Before performing various operations on rational expressions, always ensure that the rational expression has been simplified. Look for factors common to both the numerator and the denominator to simplify a rational expression. Use the following procedure to simplify them.
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Factor the polynomials in the numerator and the denominator.
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Establish the restrictions.
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Simplify the common factors of the two polynomials.
The following operations can then be performed.
Mastering the different factoring techniques is a key element to successfully simplifying rational expressions.
It is impossible to divide by |0,| therefore, it is necessary to add restrictions in a rational expression so the denominator is not equal to |0.| Division by |0| does not exist. It is an undefined value.
The restrictions must be established before simplifying the regular expression.
Therefore, we must identify the possible variable values of the denominator’s polynomial that would create a denominator with a value of |0.|
Consider the following rational fraction.
|\dfrac{x-7}{x-3}|
It is irreducible, so we can only establish the restrictions, i.e., the values of |x| where the denominator cancels itself.
|x – 3| is the denominator.
Add the restriction.
|x - 3 \neq 0|
|x \neq 3|
Restrictions: The value |3| cannot be assigned to the variable |x,| since the fraction would have an undefined value.
Consider the following rational fraction.
||\dfrac{x^{2}+10x+25}{x^{3}+5x^{2}}||
The trinomial can be factored by the numerator using the product-sum technique.
|\dfrac{x^{2}+5x+5x+25}{x^{3}+5x^{2}}|
|=\dfrac{x(x+5)+5(x+5)}{x^{3}+5x^{2}}|
|=\dfrac{(x+5)(x+5)}{x^{3}+5x^{2}}|
Remove a GCF from the denominator.
|\dfrac{(x+5)(x+5)}{x^{2}(x+5)}|
Add the restrictions.
In the following, there are two restrictions, because of the two factors in the denominator.
|x^2 \neq 0| thus, |x \neq 0|
|x + 5 \neq 0| thus, |x \neq -5|
There is the term |x+5| above and below, so it can be simplified, resulting in:
|\dfrac{(x+5)}{x^{2}}| where |x \neq -5| and |x \neq 0|
Restrictions: The values |-5| and |0| cannot be assigned to the variable |x,| since the fraction would have an undefined value.