Decode Expressions: Terms, Factors & Coefficients

As shown in Figure 1, terms are separated by $+$ or $-$, while factors are things being multiplied.

Example 1 (Straightforward)

Problem: In $7x + 3$, identify the terms and the coefficient of $x$.

Step 1: Split by addition: the terms are $7x$ and $3$. (Terms are the "chunks" added/subtracted.)

Step 2: The coefficient of $x$ is the number multiplying $x$. In $7x$, that number is 7.

Key insight: Coefficients attach to variables through multiplication.

Why wrong answers happen: Saying the coefficient is $x$ confuses variable with coefficient; saying it's $3$ grabs the constant term.

Transferable process: First find terms (split by $+$ / $-$), then find coefficients inside variable terms.

Example 2 (Moderate)

Problem: For $-4a^2b + 6b - 9$, list the terms and the factors of the first term.

Step 1: Identify terms: $-4a^2b$, $6b$, and $-9$.

Step 2: Factor the first term: $-4a^2b = (-4)\cdot a\cdot a\cdot b$. (Exponents mean repeated factors.)

Key insight: A negative sign belongs to the term; treat it as part of the coefficient.

Why wrong answers happen: Listing $4a^2b$ as a term ignores the minus; listing factors as $-4a^2$ and $b$ is incomplete (you can, but be consistent about what counts as a factor).

Transferable process: Keep the sign with the term; expand exponents into repeated factors when asked.

Test Tip: If the question asks for terms, look only for $+$ / $-$ splits. If it asks for factors, look for multiplication (including implied multiplication like $3x$).

Example 3 (Challenging)

Problem: In $2(x-3) + 5x(x+1)$, how many terms are in the expression after simplifying?

Step 1: Distribute: $2(x-3)=2x-6$.

Step 2: Multiply: $5x(x+1)=5x^2+5x$.

Step 3: Combine like terms: $2x-6+5x^2+5x = 5x^2+7x-6$.

Answer: There are 3 terms: $5x^2$, $7x$, and $-6$.

Why wrong answers happen: Counting terms before simplifying (seeing $2(x-3)$ as "one term") or forgetting to combine $2x$ and $5x$.

Transferable process: When asked "after simplifying," distribute/multiply first, then combine like terms, then count terms.