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Section 8.4 Closing Ideas

Factoring is a basic but important algebraic manipulation. It's nothing more than the distributive property (Definition 3.1.4 ) applied backwards.

\begin{equation*} \begin{aligned} 3x + 4x \amp = (3 + 4)x = 7x \amp \eqnspacer \amp \text{Combine like terms} \\ x^2 + 4x \amp = (x + 4)x \amp \amp \text{Factor out the common factor} \end{aligned} \end{equation*}

And if there were some aspects of this section that feels familiar, it's because this is the exact same algebra that is used with combining like terms.

\begin{equation*} \begin{aligned} 3x + 4x \amp = (3 + 4)x = 7x \amp \amp \text{Combine like terms} \\ x^2 + 4x \amp = (x + 4)x \amp \amp \text{Factor out the common factor} \end{aligned} \end{equation*}

The difference between combining like terms and factoring out the common factor is that there's an extra step of arithmetic that we can do with numbers that we can't do with algebraic expressions.

Math is an extremely scaffolded subject. What this means is that new ideas are very often built on older ones. This also means that weaknesses in the foundation make the higher levels of mathematical thinking less stable. Hopefully, as you've been working your way through these sections, you have been taking the time to think through the ideas and solidify those core concepts.