Dividing Rational Expressions Step-by-Step

Let's understand what rational expressions are and how to set up division problems.A rational expression is any fraction that contains variables. Here are some examples.Before we work with variables, let's recall how we divide regular fractions.This same principle applies when we have variables. Let's look at a more complex example.When dividing rational expressions, we first write the problem clearly, then change division to multiplication and flip the second fraction.Let's break down these steps to make sure we understand the process completely.Before we move on to factoring and simplifying, here are some important points to remember.Now that we understand how to set up the division problem correctly, we're ready to move on to factoring the expressions.When factoring rational expressions, we need to factor both numerators and denominators completely.Let's start by reviewing different factoring patterns. First, always look for common factors.Here, we can factor out six x y, leaving us with two x plus three y in parentheses.Next, let's look at factoring trinomials. When factoring x squared plus five x plus six...We find factors that multiply to give six and add to give five, resulting in x plus two times x plus three.For a difference of squares like x squared minus four...We factor it as x plus two times x minus two. Remember, this pattern only works for perfect squares.Now, let's apply these patterns to our main expression. We'll factor each part completely.Here are some important tips to remember when factoring rational expressions.Let's also review some common mistakes to avoid when factoring.Let's look at our original expression one more time, now fully factored. Notice how each part has been factored completely.When working with rational expressions, we must identify all values that make any denominator equal to zero.Let's examine our first denominator. When x minus 2 equals zero, x equals 2.For our second denominator, x squared minus 4 equals zero. This factors to x plus 2 times x minus 2.Solving this gives us x equals negative 2 or x equals 2.Therefore, our domain restrictions are x cannot equal 2 and x cannot equal negative 2.We can visualize these restrictions on a number line. The red dots show the excluded values.After converting division to multiplication, we must check if any new restrictions appear.In this case, no new restrictions appear because all denominators were present in the original expression.Our final domain restriction remains: x cannot equal 2 or negative 2.Now that we've identified our domain restrictions, we can proceed with canceling common factors.After canceling common factors, we need to multiply the remaining terms.Let's multiply the numerators and denominators separately to keep our work organized.In the numerator, we multiply x plus 2 and x plus 5, giving us x squared plus 7x plus 10.In the denominator, we multiply x plus 3 and x plus 4, giving us x squared plus 7x plus 12.Our simplified rational expression becomes x squared plus 7x plus 10 over x squared plus 7x plus 12.Let's verify our domain restrictions are complete and correct.Finally, let's go through our checklist to ensure our answer is complete.Let's review the key points about simplifying rational expressions.Thanks for learning about simplifying rational expressions with Spark.E!