Skip to main content

Section 19.5 Going Deeper: Additional Factoring Techniques

In Section 9.1 and Section 9.5, we looked at various factoring ideas and techniques. We're going to push that a little bit deeper here. Before getting started, we want to remind ourselves of the most common special factorizations patterns from that section:

\begin{equation*} \begin{array}{ll} \text{Square of a binomial sum:} \amp a^2 + 2ab + b^2 = (a + b)^2 \\ \text{Square of a binomial difference:} \amp a^2 - 2ab + b^2 = (a - b)^2 \\ \text{Difference of squares:} \amp a^2 - b^2 = (a + b)(a - b) \\ \text{Sum of cubes:} \amp a^3 + b^3 = (a + b)(a^2 - ab + b^2) \\ \text{Difference of cubes:} \amp a^3 - b^3 = (a - b)(a^2 + ab + b^2) \end{array} \end{equation*}

When working with these expressions, we noted how we can use these formulas to factor expressions even when the expressions contain multiple variables or higher powers of the variables. As long as the expression can be made to fit the form, then the factorization works.

But there's something else that can happen with those more complicated expressions, which is that it may take multiple factorizations to factor it completely. In other words, after one factorization step, you may discover that the factors can be factored again. Here is an example:

\begin{equation*} \begin{aligned} x^4 - 1 \amp = (x^2 + 1)(x^2 - 1) \amp \eqnspacer \amp \text{Difference of squares} \\ \amp = (x^2 + 1)(x + 1)(x - 1) \amp \amp \text{Difference of squares} \end{aligned} \end{equation*}

Unfortunately, there is no indicator that you will need to continue to factor other than recognizing that you can continue to factor. However, as you gain experience, you will get better at recognizing when expressions can be factored and when they can't.

It turns out that the \(ac\) method of factoring can also be extended to factor certain types of expressions. Expressions of this type are said to be quadratic in form, which means that we can treat them as a quadratic expression by thinking about them in the right way. Here is an example:

\begin{equation*} x^4 + 4x^2 - 5 \end{equation*}

This is not a quadratic expression, but it is quadratic in form. This means that if we think about it the right way, we can treat it like a quadratic expression. Specifically, we can rewrite it by treating \(x^2\) as the variable:

\begin{equation*} \begin{aligned} x^4 + 4x^2 - 5 \amp = (x^2)^2 + 4(x^2) - 5 \amp \eqnspacer \amp \text{Rewrite as a quadratic} \\ \amp = (x^2 + 5)(x^2 - 1) \amp \amp \text{$ac$ method} \\ \amp = (x^2 + 5)(x + 1)(x - 1) \amp \amp \text{Difference of squares} \end{aligned} \end{equation*}

It would be nice if there were some fixed set of rules for when to factor expressions and when to leave them alone. Unfortunately, those decisions are often driven by context. The most common place this becomes an issue is with a difference of squares. For example, consider this quadratic expression:

\begin{equation*} x^2 - 3 \end{equation*}

On the one hand, we can look at this and say that is not a perfect square and just leave it in that form. On the other hand, while isn't a perfect square, it is the square of which means that we can still factor it.

\begin{equation*} \begin{aligned} x^2 - 3 \amp = x^2 - (\sqrt{3})^2 \amp \eqnspacer \amp \text{Rewrite $3$ as a square} \\ \amp = (x + \sqrt{3})(x - \sqrt{3}) \amp \amp \text{Difference of squares} \end{aligned} \end{equation*}

Does this mean that we should factor \(x^2 - 3\) using square roots? The answer is that it depends. The factorization into linear terms is helpful for some mathematical operations, but other times it's an unnecessary step. So the decision of whether to factor will be dependent upon the context. This is an idea that we've touched on many times before, which is that you don't want to think about math as a set of rules that you follow every single time. It is better to recognize that you have the option of taking that factorization one step further, but then to decide what you're going to do based on the specific goals for the problem.

This may not be the last factoring technique that you'll encounter. In many precalculus courses, you will learn about other techniques that can be applied to help to sometimes factor higher degree polynomials that don't fit any of our current ideas. But even with that technique, you still will not be able to factor everything. In fact, one of the big questions that mathematicians had wondered about for a long time is whether there's a formula that can be used to factor everything. This question was ultimately answered in the early 1800s, when it was proven that there are fifth degree equations that have roots that are "impossible" write down.