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Section 9.5 Going Deeper: Special Factorizations

There are certain factoring patterns that are useful to learn to recognize. It's not that you would be unable to factor these without recognizing them, but they come up so frequently that they have special names so that we can identify them when they happen:

The basic idea of using these patterns to factor is that you need to match your expression with one of the above formulas and identify the appropriate values of \(a\) and \(b\text{.}\) For example, suppose you want to factor \(x^2 + 6x + 9\) and you want to check whether this fits one of the patterns above. By counting the number of terms and looking at the signs, we can see that there's only one formula that has a chance.

Once we know that we should be focusing on the square of a binomial sum, we can try to identify the \(a\) and the \(b\) by looking at the terms on the end and then check to see if the term in the middle matches. We can see that we can take \(a = x\) and \(b = 3\text{,}\) which means that the middle term should be \(2ab = 6x\text{.}\) And this all matches perfectly.

\begin{equation*} x^2 + 6x + 9 = x^2 + 2 \cdot x \cdot 3 + 3^2 = (x + 3)^2 \end{equation*}

The nice part about these formulas is that it gives us insights into other factorizations that we may not immediately see using the method. Here's an example:

\begin{equation*} x^2 + 2 \sqrt{2} x + 2 = x^2 + 2 \cdot x \cdot \sqrt{2} + ( \sqrt{2})^2 = (x + \sqrt{2})^2 \end{equation*}

It's not that the method can't be applied here. Here's how it would look.

\begin{equation*} x^2 + 2 \sqrt{2} x + 2 \longrightarrow \left\{ \begin{array}{l} \text{Multiply to $2\sqrt{2}$} \\ \text{Add to $2$} \end{array} \right. \end{equation*}

It's technically true that \(\sqrt{2} + \sqrt{2} = 2 \sqrt{2}\) and \(\sqrt{2} \cdot \sqrt{2} = 2\text{,}\) and we would be able to do the factorization by grouping:

\begin{equation*} \begin{aligned} x^2 + 2 \sqrt{2} x + 2 \amp = x^2 + \sqrt{2} x + \sqrt{2} x + 2 \amp \eqnspacer \amp \text{The $ac$ method} \\ \amp = x(x + \sqrt{2}) + \sqrt{2} (x + \sqrt{2}) \amp \amp \text{Factor by grouping} \\ \amp = (x + \sqrt{2})(x + \sqrt{2}) \amp \amp \text{Factor out the common factor} \end{aligned} \end{equation*}

But we can see from this that the \(ac\) method is really designed for thinking through integer factorizations. Recognizing the factorization with square roots is just not something we should expect students at this level to do.

The reason we use these special factorizations is because they are a special pattern that we can learn to recognize with practice. And that's pretty much all this is. By learning to recognize these patterns, you have access to a larger collection of factorizations that you might otherwise miss. Here are some examples:

Another aspect of these formulas is that your awareness of them can help you to avoid a few very specific algebraic errors. If you are familiar with the formulas for the square of a binomial sum and difference, you would immediately recognize what these expressions should be. These are errors that are so common that one of them has been given the nickname of "the freshman's dream."

You might have noticed that that there's a formula for the square of a binomial sum and difference, but only a formula for the difference of squares and not a sum of squares. It turns out that the sum of squares cannot be factored using the tools that we have developed so far. We can use logic to prove that we can't do this. Let's try to factor \(x^2 + 9\) as an example. Using the method, we have to find values that do the following:

\begin{equation*} x^2 + 9 \longrightarrow \left\{ \begin{array}{l} \text{Multiply to $9$} \\ \text{Add to $0$} \end{array} \right. \end{equation*}

In order to get two numbers to multiply to a positive number, they must either both be positive or both be negative. But if they are both positive or both negative, then when you add them together, you can't get zero because both numbers have the same sign. And so there aren't any numbers that will do this for us.

The special factorization formulas don't stop there. Factorization is such an important concept in mathematics that there are all sorts of factorizations that can be helpful at various times and various situations. Here are a few examples, all of which you can check by multiplying everything out:

  • The sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

  • The difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

  • The difference of fourth powers: \(a^4 - b^4 = (a - b)(a^3 + a^2b + ab^2 + b^3)\)

  • A "magic" formula: \(x^2 + bx + c = \left( x - \left( - \frac{b}{2} + \sqrt{\frac{b^2}{4} - c} \right) \right) \left( x + \left( - \frac{b}{2} - \sqrt{\frac{b^2}{4} - c} \right) \right)\)

The last formula is not one that you're likely to find anywhere, but it may look similar to something you might remember from your past mathematical experiences. And it's actually related to an approach to factoring that was known by mathematicians over 3500 years ago that was introduced to the modern world by Dr. Po-Shen Loh in 2019.