Decode Expressions: Terms, Factors & Coefficients

• Core idea (what matters most): An expression is built from terms (pieces separated by $+$ or $-$), and each term is built from factors (things multiplied). A coefficient is the number factor attached to a variable factor. Example: in $-3x^2y$, the coefficient is $-3$, and the factors are $-3$, $x^2$, and $y$.

• How to apply it (method): First, circle the plus/minus signs that separate terms (like splitting $3x^2 + 5x - 7$ into three terms). Then, inside each term, break multiplication into factors (including parentheses): in $5(x-2)y$, the factors are $5$, $(x-2)$, and $y$. To find the coefficient, look for the numeric multiplier of the variable part (remember: $x$ means $1x$, and $-x$ means $-1x$).

• Common pitfalls to avoid: Don't treat subtraction as multiplication—$x-2$ is one term (two terms inside, but the whole parenthesis is a single factor when multiplied). Don't miss hidden 1s and negative signs: $-(x+3)$ has coefficient $-1$ on the factor $(x+3)$. Exponents don't create new terms: $3x^2$ is still one term.

• Quick test-day strategy (fast + accurate): Use the "Split, then Factor" rule: (1) split into terms by $+$ / $-$, (2) factor each term by multiplication, (3) name the coefficient as the number factor. If the question asks for "a factor," parentheses like $(x-2)$ absolutely count.

Quick check — Before you answer, verify…

• Did I split terms only at top-level $+$ / $-$ (not inside parentheses)?
• Did I include all factors (numbers, variables, parentheses) for the chosen term?
• Did I handle signs and hidden coefficients ($1$, $-1$) correctly?
• Am I mixing up "term" (add/subtract) with "factor" (multiply)?

As with combining like terms, structure comes first: identify the pieces correctly, then do the math. With a quick split-and-factor scan, you can nail these problems confidently.