First, you are asked to memorize set-in-stone rules, properties, or theorems.
Then, you are expected to see when and how to use them accurately. I call this algebraic intuition.
The issue: Problems can be extremely complicated and it's not so obvious which rule can be used when.
I will show you the algebraic intuition in precalculus problems so:
You will be able to point out which rule you are allowed to use instead of guessing.
You will use the rule correctly.
Let's visit one of our old friends: terms and factors.
Terms and Factors
Term, Factor Terms are entities separated by subtraction and addition. Factors are entities separated by multiplication.
For each entity, identify if they are comprised of terms or factors.
$x$
$x$ is both a term and a factor.
It's a term because $x = x + 0$ and it's a factor because $x = x \cdot 1$.
$x - 2$
$x$ and $2$ are separated by subtraction, so terms.
$(x - 2)$
We took the terms $x - 2$ and put parentheses around them. So $(x - 2)$ is a factor because $(x-2) = (x - 2) \cdot 1$.
This example shows us a fundamental rule: grouping two or more terms together in parentheses creates a factor.
$(x-2)(x+4)$
There is a multiplication outside the parentheses. So one interpretation is factors: \[\underbrace{(x-2)}_{\text{factor}}\underbrace{(x+4)}_{\text{factor}}\]
But we can also zoom into each parentheses. Doing so gives terms:
\[(\underbrace{x}_{\text{term}}-\underbrace{2}_{\text{term}})(\underbrace{x}_{\text{term}}+\underbrace{4}_{\text{term}})\]
But when is an entity a term or a factor? Which interpretation do we use?
The last example shows us a fundamental law:
Using the word "term" or "factor" requires you to specify the context, or the environment in which an entity is a term/factor.
Global context, local context Global context refers to the context of the entire expression. Local context refers to a context smaller than the entire expression.
Two important facts:
In each context level, only choose terms or factors. Not both!
Terms and factors are context creators! Zoom in on one term or factor to increase one level in context.
List all possible terms/factors (and their contexts) for the expression \[(x-2)(x+4) + 3\]
Solution: $\times$ and $+$ are present outside the parentheses. We choose $+$ first: terms.
If you zoom correctly, the expression will always swap between terms and factors.
In practice, you only need to differentiate between global and local context.
Terms Take Priority Over Factors
In the previous example, the global context had terms. Why not factors first?
Consider the expression $2 \cdot 3 + 5$. When using PEMDAS, we do \[2 \cdot 3 + 5 = 6 + 5 = 11\]
Because factors (multiplication) disappears before terms (addition), terms always takes priority in structure. The addition structure (terms) stays until the end of the calculation!
In the expression \[(2x+3)(2x-3) - (x+2)4\] is $(2x-3)$ a global or a local factor?
$(2x + 3)(2x - 3)$ and $(x + 2)4$ are global terms. Zooming into the term $(2x + 3)(2x - 3)$, we can now say $(2x - 3)$ is a factor.
Therefore $(2x - 3)$ is a local factor.
Dealing With Fractions
For fractional expressions, the numerator and denominator are separate global contexts.
Identify if you have terms or factors in the global context.
$\dfrac{(x^2 + 1)(x + 3) + 4}{x + 3}$
Both numerator and denominator have global terms:
\[\dfrac{\overbrace{(x^2 + 1)(x + 3)}^{\text{term}} + \overbrace{4}^{\text{term}}}{\underbrace{x}_{\text{term}} + \underbrace{3}_{\text{term}}}\]
$\dfrac{-x(x-2)(2x+3)}{2x + 3}$
Numerator has global factors, denominator has global terms:
\[\dfrac{\overbrace{-x}^{\text{factor}}\overbrace{(x-2)}^{\text{factor}}\overbrace{(2x+3)}^{\text{factor}}}{\underbrace{2x}_{\text{term}} + \underbrace{3}_{\text{term}}}\]
Most mistakes come from failing to identify terms/factors and their context level.