If you know how to add or subtract fractions with the same or different denominators, adding and subtracting rational expressions should be easy for you. The procedures between the two are very similar. Show
Let’s review by going over two examples: one with the same denominator, and another with different denominators.
Steps on How to Add and Subtract Rational Expressions1) Make the denominators of the rational expressions the same by finding the Least Common Denominator (LCD). Note: The Least Common Denominator is the same as the Least Common Multiple (LCM) of the given denominators. 2) Next, combine the numerators by the indicated operations (add and/or subtract) then copy the common denominator. Note: Don’t forget to simplify further the rational expression by canceling common factors, if possible. As they say, practice makes perfect. So we will go over six (6) worked examples in this lesson to illustrate how it is being done. Let’s get started! Examples of Adding and Subtracting Rational ExpressionsExample 1: Add and subtract the rational expressions below. In this case, we are adding and subtracting rational expressions with unlike denominators. Our goal is to make them all the same. Since I have monomials in the denominators, the LCD can be obtained by simply taking the Least Common Multiple of the coefficients, where LCM (3,6) = 6, and multiply that to the variable x with the highest exponent. The LCD should be (LCM of coefficients) times (LCM of variable x) which gives us \left( 6 \right)\left( {{x^2}} \right) = 6{x^2}.
The “blue fractions” are the appropriate multipliers to do the job!
Now that we have the same denominators, it is easy to simplify.
Combine similar terms (see the x variables?).
The reason is that you may have common factors, which can be canceled out.
To make this a better answer, I will exclude the value of x that can make the original rational expression undefined. I can add the condition that x \ne 0. Example 2: Add the rational expressions below. This problem contains like denominators. We want this because it is the LCD itself – the given denominator of the rational expression. So then the LCD that we are going to use is 2x + 1.
Tip: Don’t rush by immediately doing all the calculations in your head. I suggest that you place each term inside the parenthesis before performing the required operation. This extra step may be your lifesaver to avoid careless mistakes.
Do you see how I decided to place the like terms side-by-side on the numerator?
To prevent the original rational expression to have a denominator of zero, we say that x \ne - {1 \over 2}. Example 3: Add the rational expressions below. This time I have the same trinomial in both denominators. This is similar to problem #2 but the quadratic trinomial adds a layer of fun. Later, I can factor out the denominator to see if there are common factors to cancel against the numerator.
You may say that x \ne - \,4 and x \ne + \,5 from the original denominator. Example 4: Subtract the rational expressions below. This is a good example because the denominators are different. I need to find the LCD by doing the following steps. Factor each denominator completely, and line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor.
The first denominator is okay but the second one is lacking \left( {x - 5} \right). This is why I multiply it by the blue fraction.
Group similar terms together before simplifying them.
Example 5: Subtract and add the rational expressions below. This problem is definitely interesting. To solve this, hold on to the things that you already know. Find the LCD by doing the steps below. Factor each denominator completely and neatly line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor.
Example 6: Subtract and add the rational expressions below. This is our last example in this lesson. I must say this is very similar to example 5. By now, you should already have a solid understanding of how to add and subtract rational expressions. Let’s start finding the LCD again. Factor each denominator completely and neatly line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor.
Factor the denominator of the third rational equation completely.
Proceed by factoring the numerator.
You might also be interested in: Solving Rational Equations Multiplying Rational Expressions Solving Rational Inequalities How do you add or subtract rational expressions?To add or subtract two rational expressions with the same denominator, we simply add or subtract the numerators and write the result over the common denominator. When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator.
Which of the following should be determined when adding and subtracting rational expressions with different denominators?When we add or subtract rational expressions with unlike denominators, we will need to get common denominators.
How is adding and subtracting rational expressions similar to adding subtracting fractions?We can add and subtract rational expressions in much the same way as we add and subtract numerical fractions. To add or subtract two numerical fractions with the same denominator, we simply add or subtract the numerators, and write the result over the common denominator.
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