Decode Expressions: Terms, Factors & Coefficients

As shown in Figure 1 below, keep this quick reference in mind before you start a problem.

1) Plus/Minus Split (Find the Terms Fast)

When to use: When the question asks, "How many terms?" or "What are the terms in the expression?"

1. Rewrite subtraction as adding a negative (e.g., $x-3 \to x+(-3)$).
2. Split the expression at top-level $+$ signs (don't split inside parentheses).
3. Each piece is a term.

Example: $4x^2-3x+7 \to 4x^2+(-3x)+7$ has 3 terms: $4x^2$, $-3x$, $7$.

Efficiency: Saves ~20–40 seconds by preventing "splitting" mistakes like treating $3(x+2)$ as two terms.

2) Multiply Means Factor (Circle the Product)

When to use: When the question asks for "factors" or "factor of a term."

1. Look for multiplication: numbers next to variables, parentheses next to parentheses, etc.

2. Everything being multiplied is a factor.

3. If a term is written as a product (or can be), list its factors.

Example: In $-6x^2y$, the factors are $-6$, $x^2$, and $y$ (or $-6\cdot x\cdot x\cdot y$).

Efficiency: Best for multiple-choice "Which is a factor?" questions—eliminates answers that are sums.

3) Coefficient = "Number Attached" (Include the Sign!)

When to use: When asked for "the coefficient of $x$," "the leading coefficient," or "the constant term."

1. Find the term containing the variable.

2. The coefficient is the numerical factor multiplying it.

3. Keep the sign with the number.

Example: In $-3x+5$, the coefficient of $x$ is $-3$ (not 3).

Efficiency: Saves ~10–20 seconds and avoids the most common trap: dropping negatives.

4) GCF Peel-Off (Reveal Hidden Coefficients/Facts)

When to use: When a variable is "hidden" inside parentheses and the question asks for coefficients or factors.

1. Factor out the GCF of the expression or term.

2. Read coefficients/factors from the factored form.

3. Re-check by distributing quickly.

Example: $8x+12 = 4(2x+3)$. Factors include 4 and $(2x+3)$; coefficient of $x$ in the original is 8.

Efficiency: Best for expressions like $k(ax+b)$; saves ~30 seconds of guesswork.