Middle School Math Quiz: Combining Like Terms

9 questions · exam conditions

0:00

Combining Like TermsQuestion 1 of 9

Marcus is simplifying the expression $3x^2 - 7xy + 4y^2 + 2x^2 + xy - 3y^2$ . After combining like terms, he claims his result has exactly 2 terms. Which statement best explains whether Marcus is correct?

Marcus is correct because there are 2 different variables in the expression

Marcus is incorrect because the simplified expression has 3 terms:

5x^2 - 6xy + y^2

Marcus is incorrect because the simplified expression has 4 terms after combining

Marcus is correct because combining eliminates all terms with different variables

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Combining Like Terms

Practice Combining Like Terms in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Combining Like Terms, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Marcus is correct because there are 2 different variables in the expression
Marcus is incorrect because the simplified expression has 3 terms: $5x^2 - 6xy + y^2$ (correct answer)
Marcus is incorrect because the simplified expression has 4 terms after combining
Marcus is correct because combining eliminates all terms with different variables

Explanation: To combine like terms, we group terms with identical variables and exponents:

(3x^2 + 2x^2) + (-7xy + xy) + (4y^2 - 3y^2) = 5x^2 - 6xy + y^2

. This simplified expression has 3 terms, not 2. Choice A is wrong because the number of variables doesn't determine the number of terms after simplification. Choice C gives an incorrect count. Choice D is wrong because we can only combine terms that are exactly alike.

Question 2

An expression contains the terms $4x^{n+1}$ , $-2x^{n+1}$ , and $3x^n$ , where $n$ is a positive integer. After combining like terms, how many terms remain in the simplified expression?

1 term, because all terms contain the variable $x$ raised to some power
2 terms, because $4x^{n+1} - 2x^{n+1} = 2x^{n+1}$ and $3x^n$ cannot combine with it (correct answer)
3 terms, because the exponents are different variables and cannot be simplified
2 terms, because $x^{n+1}$ and $x^n$ have the same base and can be combined

Explanation: Like terms must have identical variable parts, including exponents. Here,

4x^{n+1}

and

-2x^{n+1}

are like terms because they both have

x^{n+1}

, so they combine to give

2x^{n+1}

. However,

3x^n

has exponent

n

while the other terms have exponent

n+1

. Since

n \neq n+1

, the term

3x^n

cannot be combined with

2x^{n+1}

. The final result is

2x^{n+1} + 3x^n

, which has 2 terms. Choice A incorrectly assumes same base means like terms. Choice C misunderstands the combining process. Choice D incorrectly states that different exponents can be combined.

Question 3

The expression $ax^2 + bx + c + dx^2 + ex + f$ is simplified to $3x^2 - 7x + 2$ . If $a = 5$ and $b = -3$ , what is the value of $d + e$ ?

$-6$ (correct answer)
$-2$
$2$
$6$

Explanation: When we combine like terms:

(a + d)x^2 + (b + e)x + (c + f) = 3x^2 - 7x + 2

. This means

a + d = 3

b + e = -7

, and

c + f = 2

. Given

a = 5

5 + d = 3

, so

d = -2

. Given

b = -3

-3 + e = -7

, so

e = -4

. Therefore,

d + e = -2 + (-4) = -6

. Choice B gives just the value of

d

. Choice C gives

-d

. Choice D gives

|d + e|

or results from sign errors in solving for

d

and

e

Question 4

Sarah claims that $7xy^2$ and $-3x^2y$ are like terms because "they both have $x$ , $y$ , and the same total degree." Which statement best evaluates Sarah's reasoning?

Sarah is correct because both terms have degree 3 and contain the same variables
Sarah is incorrect because like terms must have identical variable parts, and $xy^2 \neq x^2y$ (correct answer)
Sarah is correct because the coefficients are different, which is what makes terms combinable
Sarah is incorrect because the terms have different coefficients, so they cannot be like terms

Explanation: For terms to be like terms, they must have exactly the same variable part, including identical exponents on each variable. In

7xy^2

, we have

x^1y^2

, while in

-3x^2y

, we have

x^2y^1

. Even though both terms contain

x

and

y

and both have total degree 3, the exponents on each variable are different, so

xy^2 \neq x^2y

. Therefore, these are not like terms and cannot be combined. Choice A incorrectly focuses on total degree. Choice C misunderstands what makes terms combinable. Choice D incorrectly suggests coefficients determine whether terms are like terms.

Question 5

An algebraic expression contains terms $5x^2y$ , $-3xy^2$ , $2x^2y$ , $4xy^2$ , and $-7x^2y$ . After combining like terms, which expression represents the result?

$-7xy^2$ because the $x^2y$ terms cannot be combined with $xy^2$ terms
$xy^2$ because all terms can be combined since they contain the same variables
$0$ because when all like terms are combined, they cancel out completely
$xy^2$ because the $x^2y$ terms sum to zero and only $xy^2$ terms remain (correct answer)

Explanation: When you encounter algebraic expressions with multiple terms, you need to identify and combine like terms—terms that have identical variable parts with the same exponents. Let's work through this systematically. First, group the terms by their variable parts: the

x^2y

terms are

5x^2y

2x^2y

, and

-7x^2y

, while the

xy^2

terms are

-3xy^2

and

4xy^2

. Now combine each group separately. For the

x^2y

terms:

5x^2y + 2x^2y + (-7x^2y) = (5 + 2 - 7)x^2y = 0x^2y = 0

. For the

xy^2

terms:

-3xy^2 + 4xy^2 = (-3 + 4)xy^2 = 1xy^2 = xy^2

. The final result is

0 + xy^2 = xy^2

. Looking at the wrong answers: Choice A correctly identifies that

x^2y

and

xy^2

terms cannot be combined, but incorrectly states the final answer. Choice B makes the fundamental error of assuming all terms can be combined just because they contain the same variables—the exponents must also match exactly. Choice C incorrectly concludes that everything cancels to zero, missing that the

xy^2

terms don't cancel out. Choice D correctly identifies that the

x^2y

terms sum to zero while the

xy^2

terms remain as

xy^2

. Study tip: Always separate terms by their complete variable part (including exponents) before combining. Terms like

x^2y

and

xy^2

are completely different because the exponents are distributed differently between the variables.

Question 6

The expression $3a^2b - 2ab^2 + 5ab + a^2b - 4ab + ab^2$ is simplified by combining like terms. How many different types of terms (distinct variable parts) are present in the original expression?

4 types: $a^2b$ , $ab^2$ , $ab$ , and constant terms that will appear after simplification
2 types: terms with $a^2$ and terms with $b^2$ , since these are the highest powers
6 types: each term has a different coefficient, making them all different types
3 types: terms with $a^2b$ , terms with $ab^2$ , and terms with $ab$ (correct answer)

Explanation: When you encounter algebraic expressions with multiple terms, identifying like terms means looking at the variable parts only—coefficients don't matter for classification. Like terms have exactly the same variables raised to exactly the same powers. Let's examine each term's variable part in this expression:

3a^2b

has variable part

a^2b

-2ab^2

has

ab^2

5ab

has

ab

a^2b

has

a^2b

-4ab

has

ab

, and

ab^2

has

ab^2

. Notice that some variable parts repeat—this is what allows us to combine like terms. Counting the distinct variable parts, we find three types:

a^2b

ab^2

, and

ab

. The first type appears in two terms, the second type appears in two terms, and the third type appears in two terms. Choice A incorrectly includes "constant terms," but there are no constant terms in this expression—every term contains variables. Choice B misunderstands the question by focusing on the highest powers of individual variables rather than the complete variable parts of each term. Choice C confuses coefficients with term types; different coefficients don't create different types of terms if the variable parts are identical. The correct answer is D: there are 3 types of terms based on their variable parts:

a^2b

ab^2

, and

ab

. Study tip: When identifying like terms, cover up the coefficients with your finger and look only at the variables and their exponents. Terms are "like" only when these variable parts match exactly.

Question 7

A student simplified $5a^2b - 3ab^2 + 2a^2b + 4ab^2 - ab^2$ and got $7a^2b + 0ab^2$ . The student then wrote the final answer as $7a^2b$ . What error, if any, did the student make?

The student incorrectly combined $a^2b$ terms; the coefficient should be $6$ instead of $7$
The student incorrectly combined $ab^2$ terms; the result should be $ab^2$ instead of $0ab^2$
The student made no error; $7a^2b$ is the correct simplified form of the expression (correct answer)
The student incorrectly assumed that $a^2b$ and $ab^2$ are like terms when they are not

Explanation: Let's check the student's work: For

a^2b

terms:

5a^2b + 2a^2b = 7a^2b

✓. For

ab^2

terms:

-3ab^2 + 4ab^2 - ab^2 = 0ab^2 = 0

✓. The student correctly identified that

a^2b

and

ab^2

are not like terms, combined each group separately, and properly wrote

7a^2b + 0 = 7a^2b

. Choice A gives wrong coefficient calculation. Choice B miscalculates the

ab^2

terms. Choice D incorrectly suggests the student confused unlike terms.

Question 8

A polynomial is written as $P(x) = 2x^3 - x^2 + 4x + 1 + kx^2 - 3x + 2$ , where $k$ is a constant. For what value of $k$ will $P(x)$ have no $x^2$ term after combining like terms?

$k = 0$
$k = -1$
$k = 1$ (correct answer)
$k = 3$

Explanation: When you encounter a polynomial with like terms that need to be combined, your goal is to group terms with the same power of

x

and simplify the expression. Let's rewrite

P(x) = 2x^3 - x^2 + 4x + 1 + kx^2 - 3x + 2

by grouping like terms:

$x^3$ terms: $2x^3$
$x^2$ terms: $-x^2 + kx^2 = (-1 + k)x^2$
$x$ terms: $4x - 3x = x$
Constant terms: $1 + 2 = 3$

P(x) = 2x^3 + (-1 + k)x^2 + x + 3

. For the

x^2

term to disappear completely, its coefficient must equal zero:

-1 + k = 0

, which means

k = 1

. Looking at the wrong answers: Choice A (

k = 0

) would give us

(-1 + 0)x^2 = -x^2

, so the

x^2

term remains. Choice B (

k = -1

) results in

(-1 + (-1))x^2 = -2x^2

, making the

x^2

term even larger. Choice D (

k = 3

) gives us

(-1 + 3)x^2 = 2x^2

, which also leaves an

x^2

term. Only choice C (

k = 1

) eliminates the

x^2

term entirely. Study tip: When asked to eliminate a term from a polynomial, set the coefficient of that term equal to zero after combining like terms. This type of question tests your ability to manipulate algebraic expressions systematically.

Question 9

Consider the expression $2(3x - 4y) + 5x - 3(2y - x)$ . When simplified by combining like terms, what is the coefficient of $y$ ?

$-14$ (correct answer)
$-2$
$-10$
$2$

Explanation: First distribute:

2(3x - 4y) + 5x - 3(2y - x) = 6x - 8y + 5x - 6y + 3x

. Now combine like terms:

x

terms:

6x + 5x + 3x = 14x

y

terms:

-8y - 6y = -14y

. So the simplified expression is

14x - 14y

, making the coefficient of

y

equal to

-14

. Choice B results from only combining

-8y

and

6y

, forgetting the negative sign on the second

y

term. Choice C comes from

-8y - 2y

if the distribution error gives

-2y

instead of

-6y

. Choice D ignores the negative signs entirely.