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Section 9.4 Closing Ideas

As mentioned earlier, factoring quadratic polynomials is viewed as an important marker of your algebraic skill level. But the skill isn't just in getting the answer. In fact, the bulk of mathematics is not about getting the answer. If you look back over everything we've done so far, you will see that there is a heavy emphasis on understanding and communicating the thought processes involved in performing these algebraic manipulations.

On the last worksheet page, instead of having you do more factorizations, you were asked to seek out patterns in the factorizations you've already done. You were only asked to identify the information you could conclude in each situation, but you were hopefully also able to discover the underlying logic. That logic goes all the way back to basic arithmetic, with ideas such as the following:

  • A positive number multiplied by a positive number is a positive number.

  • A negative number multiplied by a negative number is a positive number.

  • A positive number multiplied by a negative number is a negative number.

  • A negative number multiplied by a positive number is a negative number.

There are also ideas that perhaps we don't have "memorized" phrases for, but should make sense when you think about it.

  • A positive number plus a positive number is a positive number.

  • A negative number plus a negative number is a negative number.

  • When adding a positive and a negative number together, the sign of the result will match the sign of the number that is larger (in absolute value).

Factoring quadratic polynomials is not an end in and of itself. It is a tool that allows you to do advanced algebraic manipulations, such as solving more complicated equations. So we usually do not use it in complete isolation. There is an important property of numbers that can be a useful supplemental tool.

The combination of factoring with the zero product property allows you to take an equation with higher degree and convert it into multiple equations of lower degree. Here is an example of solving the equation \(x^2 + 3x - 4 = 0\text{:}\)

\begin{equation*} \begin{array}{ccl} \begin{aligned} x^2 + 3x - 4 \amp = 0 \\ (x-1)(x+4) \amp = 0 \end{aligned} \amp \phantom{.}\hspace{1cm}\phantom{.} \amp \begin{aligned} \\ \amp \text{Factor} \\ \end{aligned} \\ \begin{array}{rlcrl} x - 1 \amp = 0 \amp \phantom{.}\hspace{5mm} \text{or} \hspace{5mm}\phantom{.} \amp x + 4 \amp = 0 \\ x \amp = 1 \amp \amp x \amp = -4 \end{array} \amp \amp \begin{aligned} \amp \text{The zero product property} \\ \amp \text{Solve each equation} \\ \end{aligned} \end{array} \end{equation*}

Notice how the factorization was reduced to just a single step and that none of the steps for solving the equation were justified. When you get further along in your math courses, you will be expected to know how to do many of those steps for yourself, which is why we're strongly emphasizing the steps here. We are creating a foundation for you to build on in the future.

In this section, we can see the full scaffolded nature of algebraic reasoning. The skill of factoring is right in the middle of the tower. Below us, we see that everything we're doing is premised on having basic fluency with algebraic concepts such as the distributive property, and it also requires us to have enough experience with numbers to search through different combinations to find the one we want. Above us, we see that there's an entirely new set of equations that we can solve once we add in some more ideas. Every time you learn a new idea in mathematics, you should take a step back to see how it fits into the bigger picture.