This week in Pre-calculus 11 we started a new unit that is all about rational expressions. At first we learned about equivalent rational expressions and then we learned how to multiply and divide them. What stuck with me was how to multiply and divide the rational expressions.

What to know:

• Rational Expression: The quotient of two polynomials ex.
• Non-Permissible Values: Similar to restrictions, Permissible values tell you what numbers x can not be equal to. These are only written if there are variables in the denominator, this is because we need to make sure that the denominator does not equal to zero since zero can’t be the denominator of a fraction. The non-permissible values are found using the zero product law. ex. x+8 = 0, x=-8 (when x = -8, the equation is equal to zero).

Example 1:

• First what you need to do is factor each fraction so that it is easier for you to simplify the expression.
• Next, simplify the expression. First what you want to do is need to check to see if there is anything in the numerator and denominator that matches (any pairs?), if so then you would cross them off since they cancel each other out, then once you have found all of the pairs you would multiply across both the numerator and denominator.
• After finding all the pairs in the question, you would re-write the fraction and see if their is anything that can be further simplified. Note: ex. expressions similar to this wouldn’t be simplified any further; this is because though it may be tempting to get rid of both 3’s, the x in the denominator is connected to the 3 by the +, this means that you would need to have something in common with both the x and the 3 to simplify it further.
• Next, since there are variables on the bottoms of the fractions, you find the non-permissible values. To do this we take a look at the original question (or its factored form) and use the product law and make the parts that involve variables equal to zero so that we can isolate and find x, these values will tell you what x can not be equal to.

Example 2:

 

• First, since this is a fraction being divided by a fraction we need to make it easier for us to solve. To do this you need to turn it into a multiplication problem, which is done by reciprocating the second fraction in the question.
• After reciprocating the second fraction, we can treat this like a multiplication question; so therefore you would factor the question, look to see if the numerator and denominator have anything in common (pairs) which you would then cross out since they cancel each other out, and then multiply across both the numerator and denominator.
• Next you would re-write the simplified expression and see if it can be further simplified.
• Now, if there are variables that are located in the denominator you need to find the permissible values. One way that dividing rational expressions differs from multiplying them is the non-permissible values. This is because when writing the non-permissible you need to take into account that both halves of the fraction that was reciprocated were the denominator at one point, therefore, you need to find the non-permissible values of both the numerator and denominator of the fraction that you reciprocated as well as the denominator of the fraction that you didn’t reciprocate.