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How do you multiply rational expressions?

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4 Answers

Altho this is an algebra question (with three very good answers given above), I'd like to add a "test-prep" tip, especially for so-called "standardized" tests, such as the SAT:

Once you have factored one polynomial, don't start all over to factor the other one. 
begin by looking for one of the SAME terms as a factor.

Since the SAT weights all questions the same, and since there is much time pressure (e.g. 20 questions in 25 minutes), no one question should consume much more than one minute, so there will ALWAYS have to be some cancellation on SAT questions like this.

Please note that "test-prep" techniques should NOT be blindly applied in school; they can backfire badly on real tests prepared by a teacher or professor (where your work counts, almost-correct answers are a plus not a minus, and good math problems are rarely "multiple choice")!  I always warn my "test-prep" students that "I'll be giving you lots of BAD advice" -- use these "test-prep" techniques only for "standardized" tests like the SAT; don't use them on school tests!



Hi, Normandy.

When multiplying rational expressions, we want both the numerators and denominators to be written as the set of their factors, then look for common factors between the numerator and denominator that we can divide out (just like we do to reduce fractions).

I am going to "multiply" the expressions in your problem to make one fraction. (Once you have the process down, you could skip this step.)

   m (m2+2m-3)
    (m2-1) my 

Factor the polynomial expressions. 


Next we divide out the common factors. (Note that we can only do this because the fraction is written as all factors. Some students make mistakes in simplifying rational expressions by "canceling out" terms that are not multiplied.)


We are left with:

    (m+3)                          m+3 
   (m+1)y          or           y(m+1)

We generally leave the answer in the factored form.

"Rational" expressions simply means that its a fraction, so you would multiply (and simplify) the same way you multiply any fractions.... like  2/3 * 9/10   

Basically, you multiply the numerators together, then multiply the denominators together... then see if any of the factors in the numerator will "cancel" with any of the factors in the denominator.  Therefore, you have to have the numerator and denominator completely factored (in factored form).  So, its easier to just factor them before you multiply. 

Step 1:  Factor each expression completely              

m / (m+1)(m-1)  *  [(m-1)(m+3)]/my

Step 2: multiply numerators, then multiply denominators [without distributing!! Leave in factored form!]    

(m)(m-1)(m+3)  /   (m+1)(m-1)(m)(y)

Step 3:  Divide out (cancel) any common factors on top and bottom    

(m+3) / [(y)(m+1)]

Hope that helps!!

I am assuming that you have two variables in the problem, m and y.  Also, I am assuming that m2 - 1 is the denominator of the first expression and m2 + 2m - 3 is the numerator of the first expression (technically, the expressions should be in parenthesis).

I am going to begin by factoring the numerators and denominators.

m/(m2 - 1) * (m2 + 2m -3)/(m*y)     Given

m/[(m -1)(m + 1)] * [(m + 3)(m - 1)]/(m*y)    Factor the numerators and denominators

1/(m + 1) * (m + 3)/y      Canceled the like term (m - 1) from the first denominator and second numerator

                                       Canceled the like term m from the first numerator and second denominator

1 * (m + 3)/[(m + 1) * y] Combined fractions into one fraction (numerators together, denominators                                                 together)

(m + 3)/(my + y)              Distributed the 1 in the numerator and the y in the denominator