Combining Like Terms - Accuplacer Qas

Subtract the expression $4p - 2q + r$ from the sum of $(p + 3q - 2r)$ and $(5p - q + 4r)$ .
Answer: $2p + 4q + r$ This is a two-step problem. First, find the sum of $(p + 3q - 2r)$ and $(5p - q + 4r)$ .\ Sum = $(p+5p) + (3q-q) + (-2r+4r) = 6p + 2q + 2r\ Next, subtract \(4p - 2q + r$ from this sum.\ $(6p + 2q + 2r) - (4p - 2q + r) = 6p + 2q + 2r - 4p + 2q - r\ Combine like terms: \((6p-4p) + (2q+2q) + (2r-r) = 2p + 4q + r\ B is incorrect because it results from a sign error when subtracting, incorrectly calculating \(2q - (-2q) = 0$ . C is incorrect because it results from reversing the subtraction order.\ D is incorrect because it results from adding all three expressions together.
The expression $a(b-c) - b(c-a) + c(a-b)$ is equivalent to which of the following?
Answer: $2ab - 2ac$ First, distribute each term outside the parentheses to the terms inside.\ $a(b) + a(-c) - b(c) - b(-a) + c(a) + c(-b)\ This gives: \(ab - ac - bc + ab + ac - bc\ Now, group and combine like terms.\ \((ab + ab) + (-ac + ac) + (-bc - bc) = 2ab + 0 - 2bc = 2ab - 2bc$ . Wait, I made a mistake in my thought process. Let's re-calculate. $a(b-c) - b(c-a) + c(a-b) = ab - ac - bc + ab + ac - bc = 2ab - 2bc$ . Let's recheck the problem. Ah, I wrote it as $b(c-a)$ instead of $b(a-c)$ as in my scratchpad. Let's use the one from the scratchpad: $a(b-c) - b(a-c) + c(a-b)$ .\ $ab - ac - (ba - bc) + (ca - cb) = ab - ac - ab + bc + ac - bc\ Combine like terms: \((ab - ab) + (-ac + ac) + (bc - bc) = 0 + 0 + 0 = 0$ . The original idea was better. Let's adjust the question to have a non-zero answer, which is more robust. Original stem: $a(b-c) - b(c-a) + c(a-b)$ .\ $ab-ac - bc+ab + ca-cb = 2ab - 2bc$ . This is a good simplification. Let's make the answer choices reflect this. The question is now 'The expression $a(b-c) - b(c-a) + c(a-b)$ is equivalent to which of the following?' and the answer is $2ab-2bc$ . But I need to re-verify that. $a(b-c) - b(c-a) + c(a-b) = ab-ac -bc+ba +ca-cb = (ab+ba) + (-ac+ca) + (-bc-cb) = 2ab - 2bc$ . Correct. So the correct answer choice is B. Let's verify my explanation is correct now. Distribute each coefficient: $ab - ac - bc + ab + ac - bc$ . Combine like terms: $(ab+ab) + (-ac+ac) + (-bc-bc) = 2ab - 2bc$ . Correct. Let's write the explanation. Distribute each variable into the parentheses: $ab - ac - (bc - ab) + (ac - bc)$ -> $ab - ac - bc + ab + ac - bc$ . Combine like terms: $(ab + ab) + (-ac + ac) + (-bc - bc) = 2ab - 2bc$ . A is what would happen if the second term was $-b(a-c)$ . C arises from a sign error, for instance $-ab-ab=-2ab$ . D arises from a different sign error. So, this is a solid question.
The expression $-4xy^2z + 6x^2yz - 3xyz^2 + 7xy^2z - 9x^2yz + 5xyz^2 - 2xy^2z$ simplifies to which of the following?
Answer: $xy^2z - 3x^2yz + 2xyz^2$ When you encounter algebraic expressions with multiple terms, your goal is to combine like terms—terms that have identical variable parts with the same exponents. Here, you need to identify and group terms with matching variable combinations. Let's organize the expression by grouping like terms: - Terms with $xy^2z$ : $-4xy^2z + 7xy^2z - 2xy^2z$ - Terms with $x^2yz$ : $6x^2yz - 9x^2yz$ - Terms with $xyz^2$ : $-3xyz^2 + 5xyz^2$ Now combine each group: - $xy^2z$ terms: $-4 + 7 - 2 = 1$ , giving us $xy^2z$ - $x^2yz$ terms: $6 - 9 = -3$ , giving us $-3x^2yz$ - $xyz^2$ terms: $-3 + 5 = 2$ , giving us $2xyz^2$ The simplified expression is $xy^2z - 3x^2yz + 2xyz^2$ , which matches choice C. Choice A has the correct $xy^2z$ and $xyz^2$ terms but shows $+3x^2yz$ instead of $-3x^2yz$ —a sign error in combining the $x^2yz$ terms. Choice B incorrectly shows $9xy^2z$ (perhaps from adding absolute values: $4 + 7 + 2 = 13$ , not 9) and has the wrong signs. Choice D has correct coefficients but wrong signs throughout, suggesting errors in tracking positive and negative terms. Study tip: Always organize like terms vertically when combining, and double-check your arithmetic with positive and negative numbers. Sign errors are the most common mistake on these problems.
An expression is simplified as follows: $7x - (3 - 2x) = 7x - 3 - 2x = 5x - 3$ . Which statement best describes the error in this simplification?
Answer: The distributive property was not applied correctly to the $-2x$ term. The original expression is $7x - (3 - 2x)$ . The negative sign outside the parentheses must be distributed to BOTH terms inside. The correct simplification is $7x - 3 - (-2x) = 7x - 3 + 2x = 9x - 3$ . The error occurred when the negative sign was only applied to the $3$ and not to the $-2x$ , which is an incorrect application of the distributive property. A is incorrect because, given the erroneous step $7x - 3 - 2x$ , the combination to $5x - 3$ is actually correct. C is incorrect as there is no constant term $7$ . D is incorrect because the expression in the parentheses, $3 - 2x$ , cannot be simplified further as the terms are not alike.
Which of the following expressions is equivalent to $8x - [3y - (2x + 5y)]$ ?
Answer: $10x + 2y$ To simplify an expression with nested parentheses, work from the inside out. First, distribute the negative sign inside the brackets.\ $8x - [3y - 2x - 5y]\ Next, combine like terms inside the brackets.\ \(8x - [-2x - 2y]\ Finally, distribute the negative sign outside the brackets to the terms inside.\ \(8x + 2x + 2y\ Combine the remaining like terms: \(10x + 2y$ . A is incorrect because it results from distributing the outer negative sign incorrectly.\ B is incorrect because it results from multiple sign errors.\ C is incorrect because it results from a sign error when distributing the negative sign to the $2y$ term in the brackets.
A triangle has side lengths represented by the expressions $3a - b$ , $4a + 2b$ , and $a - 5b$ . Which of the following expressions represents the perimeter of the triangle?
Answer: $8a - 4b$ The perimeter of a triangle is the sum of its three side lengths. To find the perimeter, add the three expressions together.\ Perimeter = $(3a - b) + (4a + 2b) + (a - 5b)\ Combine the \(a$ terms: $3a + 4a + a = 8a\ Combine the \(b$ terms: $-b + 2b - 5b = b - 5b = -4b\ The simplified expression for the perimeter is \(8a - 4b$ . A is incorrect due to a miscalculation of the $a$ terms ( $3+4+1=8$ , not 7). C is incorrect due to a miscalculation of the $b$ terms ( $-1+2-5=-4$ , not -3). D is incorrect due to sign errors when combining the $b$ terms.
Simplify the expression $x(y - 3) - y(x + 2) + 5(x+y)$ .
Answer: $2x + 3y$ First, use the distributive property for each part of the expression.\ $x(y) + x(-3) - y(x) - y(2) + 5(x) + 5(y)\ This gives: \(xy - 3x - xy - 2y + 5x + 5y\ Now, group and combine like terms.\ \((xy - xy) + (-3x + 5x) + (-2y + 5y)\ This simplifies to: \(0 + 2x + 3y = 2x + 3y\ A is incorrect due to a calculation error on the y-terms.\ B is incorrect because the \(xy$ terms cancel out.\ D is incorrect due to a sign error when combining the y-terms.
If $Q = -3r^2s + 6rs^2 - 8r^3 + 5r^2s - 2rs^2 + 3r^3 - 7rs^2$ , then the coefficient of $rs^2$ in the simplified form of $Q$ is:
Answer: $-3$ When you encounter a polynomial expression with multiple terms, your goal is to combine like terms—terms that have identical variable parts with the same exponents. This question tests your ability to identify and correctly combine like terms while tracking coefficients. To find the coefficient of $rs^2$ , you need to identify all terms containing $rs^2$ and add their coefficients. Looking at $Q = -3r^2s + 6rs^2 - 8r^3 + 5r^2s - 2rs^2 + 3r^3 - 7rs^2$ , the terms with $rs^2$ are: $6rs^2$ , $-2rs^2$ , and $-7rs^2$ . Adding the coefficients: $6 + (-2) + (-7) = 6 - 2 - 7 = -3$ . Therefore, when simplified, the coefficient of $rs^2$ is $-3$ . Answer choice A ( $15$ ) likely comes from adding all positive coefficients without considering signs or term types. Answer choice B ( $3$ ) probably results from the calculation $6 - 2 - 7$ but forgetting the negative sign, taking the absolute value of $-3$ . Answer choice C ( $-1$ ) might occur if you incorrectly combined fewer terms, perhaps missing one of the $rs^2$ terms entirely. The correct answer is D ( $-3$ ). Strategy tip: Always circle or highlight like terms before combining them. This visual organization prevents you from missing terms or accidentally combining unlike terms. Double-check that the variable parts match exactly— $r^2s$ and $rs^2$ are completely different terms that cannot be combined.
When $9w^2x^3 - 4wx^4 + 2w^3x^2 - 6w^2x^3 - 5wx^4 + 7w^3x^2 + 3w^2x^3 - 8w^3x^2$ is simplified, the coefficient of $w^3x^2$ is:
Answer: $1$ When you encounter a polynomial expression with multiple terms, your goal is to combine like terms—terms that have identical variable parts with the same exponents. To simplify this expression, you need to identify and group terms with the same variable combination. Let's organize the terms by their variable parts: Terms with $w^2x^3$ : $9w^2x^3 - 6w^2x^3 + 3w^2x^3$ Terms with $wx^4$ : $-4wx^4 - 5wx^4$ Terms with $w^3x^2$ : $2w^3x^2 + 7w^3x^2 - 8w^3x^2$ For the $w^3x^2$ terms specifically, combine the coefficients: $2 + 7 + (-8) = 9 - 8 = 1$ Therefore, the coefficient of $w^3x^2$ is $1$ , making choice B correct. Looking at the wrong answers: Choice A gives $-1$ , which you might get if you miscounted the signs or forgot one of the positive terms. Choice C gives $17$ , which could result from adding all the coefficients without considering the negative sign on the last term ( $2 + 7 + 8 = 17$ ). Choice D gives $9$ , which you'd get if you stopped after adding just the first two coefficients ( $2 + 7 = 9$ ) and forgot about the $-8w^3x^2$ term. Strategy tip: When combining like terms, write out each coefficient clearly with its sign, then add step by step. Double-check that you've identified all terms with the target variable combination—it's easy to miss terms in long expressions.
Simplify the expression $4(x + 3\sqrt{y}) - 3(2x - \sqrt{y})$ .
Answer: $-2x + 15\sqrt{y}$ Distribute the coefficients to the terms inside the parentheses, treating $x$ and $\sqrt{y}$ as bases for like terms.\ $4(x) + 4(3\sqrt{y}) - 3(2x) - 3(-\sqrt{y})\ This gives: \(4x + 12\sqrt{y} - 6x + 3\sqrt{y}\ Now, combine like terms.\ \((4x - 6x) + (12\sqrt{y} + 3\sqrt{y}) = -2x + 15\sqrt{y}\ A is incorrect due to a sign error when distributing \(-3$ to $-\sqrt{y}$ , resulting in $12\sqrt{y} - 3\sqrt{y} = 9\sqrt{y}$ . C is incorrect due to a sign error on the $x$ terms.\ D is incorrect because it results from incorrectly combining unlike terms.
The sum of three consecutive odd integers can be represented by the expression $(2n+1) + (2n+3) + (2n+5)$ . Which of the following is a simplified form of this expression?
Answer: $6n + 9$ To simplify the expression, remove the parentheses and combine like terms. The expression represents the sum of three terms.\ $2n + 1 + 2n + 3 + 2n + 5\ Combine the \(n$ terms: $2n + 2n + 2n = 6n\ Combine the constant terms: \(1 + 3 + 5 = 9\ The simplified expression is \(6n + 9$ . B is incorrect due to a calculation error with the constants.\ C is incorrect due to a calculation error with the $n$ terms.\ D is incorrect because it misinterprets the coefficients of $n$ .
Simplify the expression $\frac{1}{2}(6x - 10y) - \frac{2}{3}(6x - 9y)$ .
Answer: $-x + y$ First, distribute the fractional coefficients to the terms inside the parentheses.\ $\frac{1}{2}(6x) - \frac{1}{2}(10y) - [\frac{2}{3}(6x) - \frac{2}{3}(9y)]\ This simplifies to: \(3x - 5y - [4x - 6y]\ Next, distribute the negative sign to the terms in the bracket.\ \(3x - 5y - 4x + 6y\ Finally, combine like terms.\ \((3x - 4x) + (-5y + 6y) = -x + y\ A is incorrect because it results from a sign error when distributing the negative, leading to \(-5y - 6y = -11y$ . C is incorrect because it results from a sign error on the $x$ term.\ D is incorrect because it results from adding the two simplified expressions instead of subtracting.
The area of a larger rectangle is $10x^2 + 5x - 3$ . The area of a smaller rectangle inside it is $4x^2 - x - 2$ . Which expression represents the area of the region inside the larger rectangle but outside the smaller rectangle?
Answer: $6x^2 + 6x - 1$ To find the area of the region between the two rectangles, subtract the area of the smaller rectangle from the area of the larger rectangle.\ Area = \((10x^2 + 5x - 3) - (4x^2 - x - 2)\ Distribute the negative sign: \(10x^2 + 5x - 3 - 4x^2 + x + 2\ Combine like terms: \((10x^2 - 4x^2) + (5x + x) + (-3 + 2) = 6x^2 + 6x - 1\ A is incorrect because it results from an error combining the x-terms and the constant terms.\ B is incorrect because it results from an error combining the x-terms.\ D is incorrect because it results from adding the two areas instead of subtracting.
The expression $7pq^2 - 4p^2q + 9q^3 - 3pq^2 - 5p^2q + 2q^3 - 6pq^2$ simplifies to:
Answer: $-2pq^2 - 9p^2q + 11q^3$ When you encounter polynomial expressions with multiple terms, your goal is to combine like terms—terms that have identical variable parts with the same exponents. This requires careful organization and attention to signs. To simplify this expression, group terms with the same variable combinations: $pq^2$ terms, $p^2q$ terms, and $q^3$ terms. For $pq^2$ terms: $7pq^2 - 3pq^2 - 6pq^2 = (7 - 3 - 6)pq^2 = -2pq^2$ For $p^2q$ terms: $-4p^2q - 5p^2q = (-4 - 5)p^2q = -9p^2q$ For $q^3$ terms: $9q^3 + 2q^3 = (9 + 2)q^3 = 11q^3$ Therefore, the simplified expression is $-2pq^2 - 9p^2q + 11q^3$ . Looking at the wrong answers: Choice A gives $16pq^2$ , which suggests adding all the $pq^2$ coefficients without paying attention to signs ( $7 + 3 + 6 = 16$ ). Choice B shows $4pq^2$ , indicating a sign error where you might have calculated $7 + 3 - 6 = 4$ instead of $7 - 3 - 6$ . Choice C has $+p^2q$ instead of $-9p^2q$ , showing confusion about combining negative terms. Study tip: When combining like terms, write out the coefficients with their signs explicitly: $(+7) + (-3) + (-6)$ . This visual approach helps prevent sign errors, which are the most common mistakes in polynomial simplification problems.
Which of the following expressions is NOT equivalent to $5a - 2(3a - b)$ ?
Answer: $5a - 6a - 2b$ First, simplify the given expression: $5a - 2(3a - b) = 5a - 6a + 2b = -a + 2b$ . Now, check each option to see which one is not equivalent to $-a + 2b$ . A: $-a + 2b$ is equivalent. B: $2b - a$ is equivalent to $-a + 2b$ by the commutative property of addition. C: $a + 2(b - a) = a + 2b - 2a = -a + 2b$ . This is equivalent. D: $5a - 6a - 2b = -a - 2b$ . This is NOT equivalent to $-a + 2b$ . The sign of the (2b) term is different. This is a common error of forgetting to distribute the negative sign to the $-b$ term in the original expression.
What is the result when $3m^2n - 2mn^2 + m^2n$ is subtracted from $5mn^2 - m^2n$ ?
Answer: $-5m^2n + 7mn^2$ First, simplify the expression to be subtracted: $3m^2n - 2mn^2 + m^2n = 4m^2n - 2mn^2$ . The problem asks to subtract this from $5mn^2 - m^2n$ . The setup is: $(5mn^2 - m^2n) - (4m^2n - 2mn^2)$ .\ Distribute the negative sign: $5mn^2 - m^2n - 4m^2n + 2mn^2\ Combine like terms: \((-m^2n - 4m^2n) + (5mn^2 + 2mn^2) = -5m^2n + 7mn^2\ B is incorrect due to reversing the subtraction order.\ C is incorrect due to multiple sign errors.\ D is incorrect due to a sign error on the \(m^2n$ terms.
Start with the expression $8a - 5b$ . Add $2b - 3a$ to it. Then, subtract $a - 4b$ from the result. What is the final simplified expression?
Answer: $4a + b$ This is a multi-step problem. Step 1: Add the first two expressions. $(8a - 5b) + (2b - 3a) = (8a - 3a) + (-5b + 2b) = 5a - 3b$ . Step 2: Subtract the third expression from the result of Step 1. $(5a - 3b) - (a - 4b) = 5a - 3b - a + 4b$ . Step 3: Combine like terms. $(5a - a) + (-3b + 4b) = 4a + b$ . B is incorrect due to a sign error when subtracting the (4b) term in the final step.\ C results from a sign error when subtracting the $a$ term in the final step.\ D results from adding in the final step instead of subtracting.
The expression $4a^3b - 2a^2b^2 + 6ab^3 - 3a^3b + 5a^2b^2 - 2ab^3$ simplifies to which of the following?
Answer: $a^3b + 3a^2b^2 + 4ab^3$ Combining like terms: (4a³b - 3a³b) + (-2a²b² + 5a²b²) + (6ab³ - 2ab³) = a³b + 3a²b² + 4ab³. Choice B results from adding coefficients without considering signs. Choice C results from an error in combining the a²b² terms (-2 + 5 ≠ -3). Choice D results from an error in the sign of the a³b term.
If $P = 2m^2n - 3mn^2 + 5m^2n - 7mn^2 + 4n^3$ , then $P$ equals:
Answer: $7m^2n - 10mn^2 + 4n^3$ Combining like terms: (2m²n + 5m²n) + (-3mn² - 7mn²) + 4n³ = 7m²n - 10mn² + 4n³. Choice B results from sign errors in both the m²n and mn² terms. Choice C results from a sign error with the mn² terms (-3 - 7 ≠ +4). Choice D results from a sign error with the m²n terms (2 + 5 ≠ -3).
When the expression $8u^3v - 5u^2v^2 + 3uv^3 - 12u^3v + 7u^2v^2 - 9uv^3 + 4u^3v$ is completely simplified, it equals:
Answer: $0u^3v + 2u^2v^2 - 6uv^3$ When you encounter a polynomial expression with multiple terms, your goal is to combine like terms—terms that have exactly the same variable factors with the same exponents. To simplify this expression, group the terms by their variable parts: $u^3v$ , $u^2v^2$ , and $uv^3$ . For the $u^3v$ terms: $8u^3v - 12u^3v + 4u^3v = (8 - 12 + 4)u^3v = 0u^3v$ For the $u^2v^2$ terms: $-5u^2v^2 + 7u^2v^2 = (-5 + 7)u^2v^2 = 2u^2v^2$ For the $uv^3$ terms: $3uv^3 - 9uv^3 = (3 - 9)uv^3 = -6uv^3$ The simplified expression is $0u^3v + 2u^2v^2 - 6uv^3$ . Choice A shows the correct coefficients but writes $-12u^2v^2$ instead of $+2u^2v^2$ , likely from incorrectly combining the $u^2v^2$ terms as $-5 - 7 = -12$ instead of $-5 + 7 = 2$ . Choice B has $20u^3v$ instead of $0u^3v$ , probably from adding the absolute values of the $u^3v$ coefficients: $8 + 12 + 4 = 24$ or making a similar arithmetic error. Choice D contains multiple errors: the $u^3v$ coefficient should be 0, not 4, and the $uv^3$ coefficient should be $-6$ , not $+12$ . Study tip: Always organize polynomial simplification by grouping like terms first, then carefully track positive and negative signs when combining coefficients. Double-check your arithmetic, especially with subtraction.

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