Middle School Math Quiz: Expression Structure
9 questions · exam conditions
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Expression StructureQuestion 1 of 9

The expression −3x2y+6xy2−9xyz-3x^2y + 6xy^2 - 9xyz can be factored by removing the greatest common factor. What is the coefficient of the xy2xy^2 term after factoring out this greatest common factor?

-2
2
-6
6
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Middle School Math Quiz

Middle School Math Quiz: Expression Structure

Practice Expression Structure 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 Expression Structure, 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

The expression −3x2y+6xy2−9xyz-3x^2y + 6xy^2 - 9xyz can be factored by removing the greatest common factor. What is the coefficient of the xy2xy^2 term after factoring out this greatest common factor?

  1. -2 (correct answer)
  2. 2
  3. -6
  4. 6
Explanation: The greatest common factor is 3xy3xy: the GCF of coefficients ∣−3∣|-3|, ∣6∣|6|, and ∣−9∣|-9| is 3, and the GCF of variables is xyxy. However, since the first term is negative, we factor out −3xy-3xy to avoid a leading negative sign inside the parentheses. Factoring: −3x2y+6xy2−9xyz=−3xy(x−2y+3z)-3x^2y + 6xy^2 - 9xyz = -3xy(x - 2y + 3z). The coefficient of xy2xy^2 corresponds to the −2y-2y term, so the coefficient is −2-2. Choice B ignores the negative sign. Choice C would result from not dividing by the GCF correctly. Choice D would be the original coefficient before factoring.

Question 2

Consider the expression 3x2+7xy−2x2+5y−4xy+83x^2 + 7xy - 2x^2 + 5y - 4xy + 8. After combining like terms, how many terms in the simplified expression have a coefficient with an absolute value greater than 4?

  1. 0 terms
  2. 1 term
  3. 2 terms (correct answer)
  4. 3 terms
Explanation: First, combine like terms: 3x2−2x2=x23x^2 - 2x^2 = x^2, 7xy−4xy=3xy7xy - 4xy = 3xy, 5y5y stays the same, and 88 is the constant. The simplified expression is x2+3xy+5y+8x^2 + 3xy + 5y + 8. The coefficients are 1, 3, 5, and 8. Only 5 and 8 have absolute values greater than 4, so there are 2 terms. Choice A would result from not identifying any coefficients correctly. Choice B might come from only counting the constant term 8. Choice D would result from incorrectly counting the coefficient 1 as having absolute value greater than 4.

Question 3

The expression −2x3+5x2−3x+7-2x^3 + 5x^2 - 3x + 7 is written in standard form. If this expression is rewritten by factoring out −x-x from only the terms that contain xx, what will be the coefficient of the x2x^2 term in the factored portion?

  1. 2 (correct answer)
  2. -2
  3. 5
  4. -5
Explanation: Factoring −x-x from only the terms containing xx: −2x3+5x2−3x=−x(2x2−5x+3)-2x^3 + 5x^2 - 3x = -x(2x^2 - 5x + 3). The complete expression becomes −x(2x2−5x+3)+7-x(2x^2 - 5x + 3) + 7. In the factored portion (2x2−5x+3)(2x^2 - 5x + 3), the coefficient of x2x^2 is 2. Choice B would be incorrect sign handling. Choice C would be from not properly factoring out the negative. Choice D combines both errors.

Question 4

Consider the expression 2(x+4)+3(x−1)+52(x + 4) + 3(x - 1) + 5. Which of the following statements about the structure of this expression is correct?

  1. Before distributing, there are 3 terms, and after distributing there are 4 terms
  2. Before distributing, there are 3 terms, and after distributing there are 5 terms (correct answer)
  3. The expression contains exactly 2 factors in its original form before any distribution
  4. After simplifying completely, the coefficient of xx equals the sum of all constant terms
Explanation: Before distributing, the expression has 3 terms: 2(x+4)2(x + 4), 3(x−1)3(x - 1), and 55. After distributing: 2x+8+3x−3+52x + 8 + 3x - 3 + 5, which has 5 terms. Choice A miscounts the terms after distribution. Choice C is incorrect because there are more than 2 factors when considering the expressions in parentheses. Choice D is incorrect because after simplifying to 5x+105x + 10, the coefficient of xx is 5 and the constant term is 10, so they are equal, but this is coincidental and not generally true for such expressions.

Question 5

In the expression 4(2x+3)−5x+7(x−1)4(2x + 3) - 5x + 7(x - 1), which statement correctly identifies the relationship between the terms before any simplification?

  1. There are exactly 3 terms, each containing the variable xx as a factor
  2. There are exactly 4 terms when written in expanded form, with 2 positive and 2 negative terms
  3. There are exactly 5 terms when written in expanded form, with 3 positive and 2 negative terms (correct answer)
  4. There are exactly 6 terms when written in expanded form, with 4 positive and 2 negative terms
Explanation: Expanding the expression: 4(2x+3)−5x+7(x−1)=8x+12−5x+7x−74(2x + 3) - 5x + 7(x - 1) = 8x + 12 - 5x + 7x - 7. This gives us 5 terms: +8x+8x, +12+12, −5x-5x, +7x+7x, −7-7. The positive terms are 8x8x, 1212, and 7x7x (3 terms), and the negative terms are −5x-5x and −7-7 (2 terms). Choice A incorrectly counts terms before expansion. Choice B miscounts the total number of terms. Choice D would result from incorrectly treating each factor in the original expression as a separate term.

Question 6

In the expression 4x2−6x+9−2x(x+3)4x^2 - 6x + 9 - 2x(x + 3), which of the following correctly identifies the terms after expanding but before any combining of like terms?

  1. Four terms: 4x24x^2, −6x-6x, 99, and −2x(x+3)-2x(x + 3)
  2. Five terms: 4x24x^2, −6x-6x, 99, −2x2-2x^2, and 6x6x
  3. Five terms: 4x24x^2, −6x-6x, 99, −2x2-2x^2, and −6x-6x (correct answer)
  4. Six terms: 4x24x^2, −6x-6x, 99, −2x-2x, xx, and 33
Explanation: First, expand −2x(x+3)=−2x2−6x-2x(x + 3) = -2x^2 - 6x. The complete expanded expression is 4x2−6x+9−2x2−6x4x^2 - 6x + 9 - 2x^2 - 6x, which has 5 terms: 4x24x^2, −6x-6x, 99, −2x2-2x^2, and −6x-6x. Choice A doesn't expand the last factor. Choice B incorrectly shows +6x+6x instead of −6x-6x from the expansion. Choice D incorrectly separates the factors within the parentheses.

Question 7

Consider the expression 6x2y−4xy+8x2y−12xy+26x^2y - 4xy + 8x^2y - 12xy + 2. After combining like terms, which term has a coefficient that is a factor of all the other coefficients in the simplified expression?

  1. The x2yx^2y term
  2. The xyxy term
  3. No term has a coefficient that is a factor of all other coefficients
  4. The constant term (correct answer)
Explanation: When you see an expression with multiple terms, your first step is always to combine like terms—terms that have identical variable parts. This question then asks you to find which coefficient is a factor of all the others. Let's combine the like terms in 6x2y−4xy+8x2y−12xy+26x^2y - 4xy + 8x^2y - 12xy + 2. The x2yx^2y terms are 6x2y+8x2y=14x2y6x^2y + 8x^2y = 14x^2y. The xyxy terms are −4xy+(−12xy)=−16xy-4xy + (-12xy) = -16xy. The constant remains 22. So our simplified expression is 14x2y−16xy+214x^2y - 16xy + 2. Now we need to check which coefficient divides evenly into all the others. The coefficients are 14, -16, and 2. Let's test each term: For choice A, we check if 14 divides into -16 and 2. Since −16÷14-16 ÷ 14 doesn't give a whole number, 14 is not a factor of all coefficients. For choice B, we check if -16 divides into 14 and 2. Since 14÷(−16)14 ÷ (-16) doesn't give a whole number, -16 is not a factor of all coefficients. For choice C, this would mean no coefficient works as a universal factor. For choice D, we check if 2 divides into all coefficients: 14÷2=714 ÷ 2 = 7, −16÷2=−8-16 ÷ 2 = -8, and 2÷2=12 ÷ 2 = 1. Since 2 divides evenly into all coefficients, the constant term has the coefficient that's a factor of all others. Study tip: When looking for common factors among coefficients, start with the smallest absolute value—it's the most likely candidate to divide into larger numbers.

Question 8

In the expression 12x3−4x2+x−34\frac{1}{2}x^3 - 4x^2 + x - \frac{3}{4}, which term has the smallest coefficient when all coefficients are written as decimals?

  1. The x3x^3 term
  2. The x2x^2 term (correct answer)
  3. The xx term
  4. The constant term
Explanation: Converting to decimals: 12=0.5\frac{1}{2} = 0.5, −4=−4.0-4 = -4.0, 1=1.01 = 1.0, and −34=−0.75-\frac{3}{4} = -0.75. When comparing coefficients as numbers (not absolute values), −4-4 is the smallest since −4<−0.75<0.5<1-4 < -0.75 < 0.5 < 1. The x2x^2 term has coefficient −4-4. Choice A (0.5) and Choice C (1.0) are positive and thus larger. Choice D (-0.75) is negative but larger than -4.

Question 9

Consider the expression 3xy2+7x2y−2xy2+5x2y3xy^2 + 7x^2y - 2xy^2 + 5x^2y. When this expression is written in simplified form, which statement correctly describes the relationship between the coefficients?

  1. The coefficient of one term is exactly twice the coefficient of the other term
  2. The coefficient of one term is exactly three times the coefficient of the other term
  3. The coefficients are equal in absolute value but opposite in sign
  4. The coefficients have a ratio of 12:1 when written as positive integers (correct answer)
Explanation: Combining like terms: (3xy2−2xy2)+(7x2y+5x2y)=xy2+12x2y(3xy^2 - 2xy^2) + (7x^2y + 5x^2y) = xy^2 + 12x^2y. The coefficients are 1 and 12. The ratio is 12:1. Choice A would be true if one coefficient were 2 and the other 1, but we have 12 and 1. Choice B would be true if the coefficients were 3 and 1. Choice C would be true if the coefficients were equal in absolute value but opposite in sign, like 6 and -6.