Combining Like Terms - Pre Algebra

1. 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?
   Answer: 2 terms, because $4x^{n+1} - 2x^{n+1} = 2x^{n+1}$ and $3x^n$ cannot combine with it 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.
2. 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 $d + e$?
   Answer: $-6$ 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 $e$.
3. 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?
   Answer: $k = 1$ 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: $1 + 2 = 3$ So $P(x) = 2x^3 + (-1 + k)x^2 + x + 3$. 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.
4. 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?
   Answer: $xy^2$ because the $x^2y$ terms sum to zero and only $xy^2$ terms remain 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.
5. Consider the expression $2(3x - 4y) + 5x - 3(2y - x)$. When simplified by combining like terms, what is the coefficient of $y$?
   Answer: $-14$ 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.
6. 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?
   Answer: The student made no error; $7a^2b$ is the correct simplified form of the expression 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.
7. 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?
   Answer: Marcus is incorrect because the simplified expression has 3 terms: $5x^2 - 6xy + y^2$ 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.
8. 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?
   Answer: 3 types: terms with $a^2b$, terms with $ab^2$, and terms with $ab$ 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.
9. 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?
   Answer: Sarah is incorrect because like terms must have identical variable parts, and $xy^2 \neq x^2y$ 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.

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