Section 17.5 Going Deeper: Reducing Rational Expressions
Usually, when students think of calculus, they think of extremely elaborate calculations and complicated algebraic manipulations. And while there are certainly some aspects of the course for which that is true, it turns out that the core concepts of calculus are ideas that you can understand without needing a lot of algebra.
One of the largest hurdles students face when they get to calculus is not the calculus, but the algebra. It is very possible to get all the way to calculus without having attained algebraic fluency. As the expressions get more complex, the error rate goes up. And this, much more than the calculus, ends up holding students back.
One area of algebra that students struggle with in particular is the manipulation of fractions. This is an extremely important skill because the main idea of the first half of calculus (differential calculus) is defined as a fraction. In theory, you already have all the skills required to perform the algebraic part of these calculations. You know how to perform a substitution, multiply polynomials, and combine like terms. The remaining algebraic step is to reduce the algebraic fraction.
In all of the examples and problems in this section, you were given a fraction where all the numerator and denominator were simply products of algebraic terms. This helps to give students the practice of matching up terms and reducing correctly in these situations. But it also leads students down the path of incorrect manipulations when they do not fully understand the cancellation process.
Over the next several sections, the "Going Deeper" sections are going to focus specifically on dealing with fractions involving polynomials, which are often called rational expressions. These sections are aimed at students who are on the pathway towards calculus to help develop specific algebraic skills that are useful along that pathway.
Reducing fractions is a mathematical process built on multiplication. You should be looking for terms that are multiplicative in the numerator and the denominator. More precisely, you need to be able to factor identical terms from the numerator and the denominator. This is why we chose to explicitly write out the products before reducing in this section.
Most of the errors that come from reducing fractions incorrectly result from different forms of simply crossing off terms that look the same in the numerator and the denominator. When you look at these calculations, you will probably immediately recognize them as being wrong. But it's easy to identify errors when it's someone else's work and you're being told that these are errors. It's often more difficult when you're looking at your own work.
Subsection 17.5.1 A Collection of Errors
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Canceling addition: Do not cancel out terms when they are being added or subtracted. It must always be a multiplicative term to cancel.
\begin{equation*} \frac{x^2 + 5x + 4}{x^2 - 4x + 4} \overset{\times}{=} \frac{x^2 + 5x \, \cancel{+ \, 4}}{x^2 - 4x \, \cancel{+ \, 4}} = \frac{x^2 + 5x}{x^2 - 4x} \end{equation*} -
Partial cancellation: This is another version of the previous error. Even though the terms are being multiplied by something, that something isn't the entire remainder of the numerator and the denominator. The mistake here is canceling out part of an additive term.
\begin{equation*} \frac{3x + 1}{2x + 1} \overset{\times}{=} \frac{3 \cancel{x} + 1}{2\cancel{x} + 1} = \frac{3+1}{2+1} = \frac{4}{3} \end{equation*} -
Another partial cancellation: This one is tricky because there's a half-truth to the cancellation. We'll discuss this particular step in more detail later. The key for now is to recognize how we're still canceling out within an additive term.
\begin{equation*} \frac{3x + 1}{x} \overset{\times}{=} \frac{3 \cancel{x} + 1}{\cancel{x}} = 3 + 1 = 4 \end{equation*} -
Breaking the order of operations: The multiplication being canceled must respect the order of operations. In this example, the and the should be seen as being grouped together, so that this cancellation cannot happen.
\begin{equation*} \frac{3x + 1}{x + 1} \overset{\times}{=} \frac{3 \, \cancel{x + \, 1}}{\cancel{x + \, 1}} = 3 \end{equation*}
As you continue to move forward into the more algebraically complex world of rational expressions, it's likely that you're going to make some of these errors. The important thing is to not be discouraged. Mistakes are going to happen. The real key is how you respond to the mistakes when you make them. Students often have a reflex of "I forgot" and then quickly move on to the next thing. Unfortunately, this usually does not help them avoid that error in the future. It takes a certain amount of intentional effort and reflection to internalize the algebra.
A helpful recommendation is to keep a record of your algebraic errors. By writing them down, you can start to identify your own patterns of mistakes, which trains your brain to watch out for them in the future. It only takes a few seconds per mistake, but that's sometimes all it takes to to get your brain to start to recognize it.
Subsection 17.5.2 Factoring Out the Constant (Including the Negative)
Having gone through a list of examples that don't reduce, it's also important to discuss certain types of fractions that do reduce. One type of factorization that students sometimes overlook is to factor out a constant. This can sometimes reveal binomial terms that factor out that are not immediately obvious. Here is an example:
One particular example of this is factoring a negative from the binomial. In general, when it comes to applications involving factoring, we try to keep the coefficient of the \(x\) term positive because it helps us to more easily recognize terms that can cancel out. Here is an example of this: