Like and unlike terms — simplification: Question bank

This bank hunts the thinking mistakes, not the arithmetic. Each item below is a one-line reveal: read the prompt, say your answer out loud with a reason, then check. If your reason is right but the yes/no is wrong, you still learned something. If the yes/no is right but you had no reason — that's the trap catching you.

Everything here builds on Like and unlike terms — simplification. If a word feels shaky, refresh 2.1.01-Variables-constants-and-coefficients first.

True or false — justify

Recall

$5 x$ and $5 x^{2}$ are like terms because they share the coefficient $5$ . False ::: Being "like" depends only on the variable part, never the coefficient; $x$ and $x^{2}$ are different powers, so these are unlike.

Recall

$3 ab$ and $3 ba$ are like terms. True ::: Multiplication is commutative, so $ab = ba$ ; the variable parts are the same, only written in a different order.

Recall

$7$ and $- 4$ (two plain numbers) count as like terms. True ::: A constant is a term with variable part $x^{0} = 1$ ; both have "no letters", so they are like and combine to $3$ .

Recall Any two terms that contain the letter

$x$ can be added together. False ::: They must contain $x$ to the same power with the same other letters; $2 x$ and $2 x^{2}$ both "contain $x$ " yet are unlike.

Recall

$\frac{x}{2}$ and $3 x$ are like terms. True ::: $\frac{x}{2} = \frac{1}{2} x$ , a coefficient of $\frac{1}{2}$ on variable part $x$ ; the fraction is just the number in front, so they combine to $\frac{7}{2} x$ .

Recall

$- x$ and $x$ are unlike terms because one is negative. False ::: The minus sign is part of the coefficient ( $- 1$ vs $+ 1$ ), not the variable part; they are like and sum to $0$ .

Recall

$x^{2} y$ and $x y^{2}$ are like terms. False ::: The powers are attached to different letters ( $x$ squared here, $y$ squared there); the variable parts differ, so they are unlike.

Recall After fully simplifying, an expression can still contain unlike terms.

True ::: "Simplified" means every group of like terms is merged; leftover unlike terms like $3 x^{2} + 10 x - 4$ genuinely cannot be combined further.

Spot the error

Recall

$3 x + 5 y = 8 x y$ — what went wrong? The two numbers were added and the letters multiplied ::: $x$ and $y$ are unlike, so nothing combines; the answer stays $3 x + 5 y$ .

Recall

$4 x + 3 x = 7 x^{2}$ — what went wrong? The coefficients were added correctly, but the variable was wrongly "grown" to $x^{2}$ ::: In like-term addition the variable part is left untouched, so the answer is $7 x$ .

Recall

$2 x^{2} + 3 x^{2} = 5 x^{4}$ — what went wrong? The exponents were added as if the terms were multiplied ( $x^{2} \cdot x^{2}$ ) ::: Addition keeps the shared power, giving $5 x^{2}$ ; only multiplication would send $2 + 2$ to the exponent.

Recall

$5 x - 2 x = 7 x$ — what went wrong? The minus sign was dropped and $5 + 2$ used instead of $5 - 2$ ::: The sign belongs to its term, so it is $(5 - 2) x = 3 x$ .

Recall

$6 a + 4 = 10 a$ — what went wrong? A constant ( $4$ ) was merged into an $x$ -term as if it were like ::: $4$ has variable part $x^{0}$ while $6 a$ has variable part $a$ ; they stay apart as $6 a + 4$ .

Recall

$2 ab + 3 a = 5 ab$ — what went wrong? $ab$ and $a$ were treated as the same variable part ::: $ab$ means $a \times b$ , unlike plain $a$ ; the expression is already simplified as $2 ab + 3 a$ .

Recall "

$x$ has no coefficient, so $x + x = x$ ." — what went wrong? A blank in front secretly means coefficient $1$ , and $1 + 1 = 2$ was skipped ::: $x + x = 1 x + 1 x = 2 x$ .

Why questions

Recall

Why are we allowed to combine like terms at all — where does the rule come from? It is the 2.2.01-Distributive-property read backwards ::: $a x + b x = (a + b) x$ just factors out the shared variable part, so we only add the numbers left behind.

Recall

Why can't unlike terms be combined even though we clearly see numbers to add? There is no common factor to pull out ::: In $3 x + 5 y$ the letters differ, so the distributive step $(a + b) \cdot (common)$ has no "common" to use.

Recall

Why does the variable part stay unchanged when we combine, but the coefficient changes? We are counting how many copies of that variable part we have ::: $3 x + 5 x$ is "3 of the thing plus 5 of the thing = 8 of the thing"; the thing itself is never altered by counting it.

Recall

Why is a lone constant like $- 4$ "unlike" every $x$ -term? Its variable part is $x^{0} = 1$ , which no $x^{1}$ or $x^{2}$ term matches ::: matching powers is required, and $0 \neq = 1 \neq = 2$ .

Recall

Why do we bother writing answers in descending powers (like $3 x^{2} + 10 x - 4$ )? It is a shared convention so like terms line up and results are easy to compare ::: the maths is identical in any order, but standard form prevents "did I miss a term?" errors — handy in 4.1.01-Polynomials-introduction.

Edge cases

Recall What is

$x^{0}$ , and why does treating constants as $x^{0}$ matter here? $x^{0} = 1$ (for any $x \neq = 0$ ), so a constant is really a coefficient times $x^{0}$ ::: this is the honest reason constants only ever combine with other constants.

Recall Simplify

$3 x + (- 3 x)$ . What kind of "term" is the result? $0$ , the zero term — it vanishes entirely ::: like terms with opposite coefficients cancel; the variable disappears because you have zero copies of it.

Recall Is

$0 x^{2} + 5 x$ the same as $5 x$ ? Yes ::: a coefficient of $0$ means zero copies of $x^{2}$ , so that term is nothing and can be dropped.

Recall Does the

value of $x$ matter when deciding whether two terms are like? No ::: "like or unlike" is decided purely by the written variable part, before any number is ever plugged in — it is a rule about shape, not value.

Recall Are

$2 x y$ and $2 y x$ ever unlike, for some sneaky value of the letters? No, never ::: $x y = y x$ holds for all numbers, so their variable parts are always identical; this is why order of letters is not a real difference.

Recall When you finish combining and get an answer like

$3 x + 5 y$ , did you "fail" to simplify? No ::: $3 x + 5 y$ is the simplified form; being unable to combine unlike terms is the correct final state, not an incomplete one — the same truth carries into 3.1.01-Solving-linear-equations-one-variable.

Connections

Like and unlike terms — simplification — parent topic these traps stress-test
2.1.01-Variables-constants-and-coefficients — coefficient vs variable part, the source of most traps
2.2.01-Distributive-property — the "why" behind every legal combination
2.1.03-Addition-and-subtraction-of-algebraic-expressions — where these traps show up in longer problems