Idiot's Guides: Algebra I (2015)
Part V. Radical, Quadratic, and Rational Functions
Chapter 17. Rational Expressions
In This Chapter
· Understanding rational expressions and their domains
· Simplifying rational expressions
· Multiplying and dividing rational expressions
· Adding and subtracting rational expressions with the same denominator and with different denominators
One of the keys to arithmetic is a firm understanding of the place value system that is the structure of whole numbers. One of the keys to algebra is an understanding of polynomials, which have a structure that parallels a place value system. If we write the number 4,829 in expanded form, it’s 4·103 + 8·102 + 2·10 + 9. If we replace 10 with a variable, we have a polynomial, 4x3 + 8x2 + 2x + 9.
In arithmetic, once we have a sturdy grasp of the whole numbers, we usually move on to fractions. In this chapter and the next, we’re going to work with some algebraic fractions. We’ll look at their construction, their simplest form, and their arithmetic. Along the way, we’ll use much of what we know about polynomials, especially factoring.
Domain of a Rational Expression
The algebraic fractions we’re talking about are properly called rational expressions. Rational here, like in rational numbers, comes from the word ratio. These expressions are ratios of two polynomials, with the warning that the polynomial in the denominator cannot equal zero.
A rational expression is a quotient of two polynomials, provided that the polynomial in the denominator is not zero.
Because the term polynomial includes expressions as simple as constants, and others that have many terms, rational expressions can be as simple as or more complicated, like . Technically, all rational numbers are rational expressions and all polynomials are as well, because we can write , but we don’t usually bother with that.
You’ll sometimes hear rational expressions called algebraic fractions, but the two terms are not identical. There are other algebraic fractions that don’t quite fit the definition of rational expressions because their numerator or denominator (or both) are not polynomials. Expressions like have some properties in common with rational expressions but are often more difficult to work with, because the tactics used with polynomials can’t be used here. Don’t make the mistake of thinking every algebraic fraction is a rational expression.
The warning that the denominator cannot equal zero is attached to any fraction, because division by zero is impossible. In rational numbers, a zero denominator is easy to spot, but when working with rational expressions, a polynomial denominator can take different values for different values of the variable. Certain values, if substituted for the variable, could make the denominator polynomial equal zero.
It may not always be obvious what values of the variable create a zero denominator. Sometimes it’s easy. The denominator of is zero when x = 0, but the denominator of is not zero when x = 0. If you plug x = 0 into , you get , which is perfectly fine. Make a habit of examining each denominator, and asking what values of the variable will make that denominator equal zero.
To answer that question, you need to solve an equation. To determine which value of x will make the denominator of equal zero, solve the equation x − 3 = 0. That tells you that x − 3 = 0 is the value you cannot allow. To find the unacceptable values of the variable for the rational expression , you need to solve the equation x2 − 3x + 2 = 0. That will take a little bit more work.
There are two values that will cause the denominator of to become zero, x = 1 and x = 2. Avoid both of those.
The values of the variable that may be substituted for the variable in any expression or function form a set called the domain of the expression. For many expressions, like polynomials, the domain is all real numbers. You can replace the variable in a polynomial with any value, and nothing troublesome will happen. With rational expressions, however, there are troublesome values more often than not. The domain of is all real numbers except 1 and 2. The simplest way to show this is to tell what’s not allowed. If you see , x ≠ 1, x ≠ 2, it means that the domain cannot include 1 or 2.
The domain of a rational expression is the set of all real numbers that can be substituted for the variable without making the denominator equal to zero.
To find the domain of the rational expression , first solve x2 − 9 = 0 to find the values that must be excluded. That will tell you the domain cannot include 3 or -3. The domain of is all real numbers except 3 and -3.
To work with the rational expression , first think about its domain, which is all real numbers except anything that will make the denominator zero. When you try to solve x2 + 1 = 0, however, you will find that x2 = -1 and that has no solution in the real numbers. So is one of those unusual rational expressions whose domain is all real numbers.
Find the domain of each rational expression.
Simplifying Rational Expressions
Rational expressions are algebraic fractions, and much of what you know about fractions will translate to rational expressions. There will be some changes, some complications, but the essential ideas will be much the same.
One of the first things we learn about fractions is that, although they may actually be two different names for the same number, it’s a lot easier to work with than with . Putting a fraction in its simplest form makes both the computation and the communication easier. When working with rational expressions, you’ll also want to have the expressions in their simplest form.
Although you may think of simplifying a fraction as a process of dividing the numerator and denominator by the same number, the reasoning behind it is a little more involved. To simplify , you’re actually factoring 84 and factoring 168.
Then look for factors that appear in both and cancel them out.
To simplify a rational expression, you’ll use a similar process. First factor the numerator and factor the denominator, if they can be factored. Then find and cancel matching factors in the numerator and denominator. Here’s an example.
To simplify , first factor each polynomial.
We can see x + 2 in both the numerator and denominator, so we’ll cancel that out.
When you’re simplifying a rational expression, it’s tempting to start crossing out anything that looks alike. Remember that “canceling” means dividing the numerator and denominator by the same number or expression. To divide properly, you need to factor the numerator and denominator and cancel factors. but . Cancel factors, not terms.
When all the factors that appear in both numerator and denominator are canceled, what remains is the simplest form of the rational expression.
Determine the domain before you simplify or do any arithmetic. The domain is the set of numbers that replace the variable in the expression you’re given, without making the denominator equal zero. is undefined for both x = 0 and x = 3, even though it can be simplified to . The domain is the domain of the unsimplified version, even after simplifying.
The polynomials that make up the rational expression may not always factor, but they may still cancel. The rational expression has a numerator that cannot be factored, but the denominator can be factored. . One of the factors of the denominator is the polynomial in the numerator. You can rewrite and then cancel to get the simplest form.
Simplify each expression and give its domain.
Operations with Rational Expressions
Rational expressions, like ordinary fractions, can never have a denominator of zero, and like ordinary fractions, are easiest to deal with when they’re in simplest form. Finding the values that must be excluded from the domain to prevent a zero denominator takes a bit of work and factoring the polynomials so that we can put the expression in simplest form will take a few more steps than simplifying a regular fraction. You know how to add, subtract, multiply, and divide fractions. Now it’s time to apply and adapt that knowledge so that you can operate with rational expressions.
Multiplication and Division
The basic rule for multiplication of fractions tells you to multiply numerator by numerator and denominator by denominator and then simplify if necessary. That rule will work with rational expressions as well, but you may not want to depend on that to handle every multiplication of rational expressions.
For this problem, it’s a workable plan.
For this multiplication, it may feel less comfortable.
Have you got some scratch paper handy? Multiplying numerator by numerator and denominator by denominator is going to take a lot of work, and when that’s done, you’ll need to figure out how to factor the results in order to simplify the product.
If you’re starting to feel like this is a job you don’t want to take on, you’re not alone. Just so you can see it, here’s what that task would look like, but you don’t know how to factor those polynomials, and you aren’t expected to, and just about anyone would look at this problem and ask if there’s a better way.
The good news is that there is a better way, and you already know the heart of it. It’s very similar to the way you simplified rational expressions, and it’s a technique you’ve used in working with fractions. You’ll simplify before multiplying. You’ll factor every polynomial that can be factored, and do as much canceling as possible. Only after that will you multiply.
Let’s look at it with a simpler example first, and then we’ll come back to the monster we looked at earlier. Let’s multiply .
First, factor each of the polynomials.
Now you want to cancel a factor from one of the numerators with a factor from one of the denominators. The factors can come from the same fraction or one from each fraction, but it must be one from a numerator and one from a denominator. Cancel the x + 3 in the first numerator with the x + 3 in the second denominator, and cancel the x − 1 in the first denominator with one of the x − 1 factors in the second numerator.
Now multiply the factors that haven’t been cancelled in the numerator, and do the same with the remaining factors in the denominator. If you did all the possible canceling in the last step, no further simplifying will be needed.
Let’s try that monster problem we saw earlier, but let’s do it by factoring and canceling first.
First, factor. Notice that x2 + 5x + 3 is not factorable, but that’s okay.
The second step is to cancel.
Finally, multiply the remaining factors.
That tames the monster a bit, doesn’t it? Factoring and canceling first should be your usual approach to multiplying rational expressions. For some simpler problems, you may choose to multiply first, and simplify later, but factoring first is usually your best move.
You probably remember that the rule for dividing fractions tells us to multiply by the reciprocal of the divisor. The same will be true for dividing rational expressions. We’ll leave the first expression as it is, change to multiplication, and invert the second rational expression.
You can remember the rule for division as “keep, change, change”. Keep the first rational expression as it is, change to multiplication, and change the second rational expression to its reciprocal.
To divide , keep just as it is for now, change the operation sign to multiplication, and replace with its reciprocal, .
When you’re dividing rational expressions, change to multiplication by the reciprocal before you start to factor. If you don’t, it’s easy to get involved in the factoring and forget to invert the divisor.
Once you’ve rewritten the division problem as an equivalent multiplication problem, it is a multiplication problem, and your tactic is to factor everything you can, cancel, and then multiply.
Perform each multiplication or division.
Addition and Subtraction with Like Denominators
Denominators are names, labels that tell what sort of thing we have. Numerators tell how many of that thing we have. If we add, or subtract, groups of the same kind of thing, the number changes but the kind of thing doesn’t.
When fractions have the same denominator, like and , adding or subtracting only requires adding or subtracting the numerators and keeping the denominator as it is. and . The same is true of rational expressions. If the denominators are the same, you only need to add or subtract the numerators, and keep the denominators the same.
To add , we’ll add (2x +5) + (x − 7) to form the new numerator, but the denominator will remain 4x − 3.
Always check to see if the sum (or difference) can be simplified. Even if both fractions were in simplest form when you started, you may be able to simplify the result.
All the usual rules apply. Combine like terms and only like terms. Simplify whenever possible, but cancel factors, not terms. That means that after you add (or subtract), you may need to factor the numerator and/or the denominator to see if there’s anything to cancel. It won’t happen every time, but it does happen. Here’s an example.
For subtraction, there is another concern to keep in mind. Remember that the fraction bar acts like a set of parentheses around the numerator, so when you place a subtraction sign between two fractions, it’s important to remember that you must subtract the entire second numerator.
When asked to subtract , you have to subtract x2, subtract 4x, and subtract -7. The simplest way to remember that is to put in parentheses and write the problem as . Distribute the minus by changing all the signs of the second polynomial, and then you can add.
Through all of that, our denominator stays the same.
Add or subtract, and simplify if possible.
Addition and Subtraction with Unlike Denominators
When you need to add or subtract fractions with different denominators, you can’t plunge in and add the numerators. That would be like trying to combine apples and oranges. Different denominators let you know that you’re dealing with different kinds of things. If you have 5 apples and 3 oranges, you don’t have 8 apples or 8 oranges. You have 8 fruits. You have to find a description that fits both apples and oranges, a common denominator.
To find the common denominator for fractions, we find the lowest common multiple of the two denominators, but when we’re working with rational expressions, the common denominator isn’t immediately apparent. You can find a common denominator for and without too much work, but finding a common denominator for and is more of a challenge. So let’s take this step by step.
1. If any denominators can be factored, factor them. Do not factor numerators. Denominators may or may not factor, and some may factor and some not.
2. Multiply each denominator by any factors present in other denominators, but not already present in this denominator. Do not simplify the denominators. Leave them in factored form.
3. Multiply each numerator by the same factors by which you multiplied its denominator. Simplify the numerators.
4. For subtraction, change the signs of the second numerator and add.
5. Add the numerators by combining like terms.
6. Check to see if the numerator of the sum will factor. Specifically, check to see if any of the factors of the common denominator are factors of the numerator. Cancel if possible.
Let’s look at an example in which the denominators do factor. We’ll subtract .
1. If any denominators can be factored, factor them.
2. Multiply each denominator by any factors present in other denominators, but not already present in this denominator.
3. Multiply each numerator by the same factors.
4. For subtraction, change the signs and add.
5. Add the numerators.
6. Check to see if the numerator will factor. Cancel if possible.
You were able to factor out a common factor of 2, but nothing will cancel. You can present the result as or , or with the denominator multiplied out as , but that multiplication isn’t usually valuable enough to make the work worthwhile.
Don’t be concerned if the denominators don’t factor. You can still find a common denominator. To add , where the denominators don’t factor, just think of each denominator as a one-factor denominator. To create a common denominator, follow the other remaining steps.
That numerator does not factor any more than that, and nothing cancels, so we can stop there. The common denominator is just the product of the two denominators. You could write the answer as if you prefer.
Add or subtract, and simplify if possible.
The Least You Need to Know
· A rational expression is the quotient of two polynomials, and is defined for all real numbers that do not make the denominator equal to zero.
· Simplify rational expressions by factoring the numerator and denominator and cancel any factor that appears in both.
· Multiply rational expressions by factoring all numerators and denominators, canceling factors from either numerator with matching factors in any denominator, and then multiplying the numerators and multiplying the denominators.
· Divide rational expressions by inverting the divisor and multiplying.
· If two rational expressions have the same denominator, add or subtract them by adding or subtracting the numerators and keeping the same denominator.
· When denominators are different, factor each denominator. Multiply the numerator and denominator of each rational expression by any factors from the other denominator that are lacking. Once you have the same denominators, add or subtract the numerators.