Expression Structure - Pre Algebra

Consider the expression $3xy^2 + 7x^2y - 2xy^2 + 5x^2y$ . When this expression is written in simplified form, which statement correctly describes the relationship between the coefficients?
Answer: The coefficients have a ratio of 12:1 when written as positive integers Combining like terms: $(3xy^2 - 2xy^2) + (7x^2y + 5x^2y) = xy^2 + 12x^2y$ . 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.
Consider the expression $2(x + 4) + 3(x - 1) + 5$ . Which of the following statements about the structure of this expression is correct?
Answer: Before distributing, there are 3 terms, and after distributing there are 5 terms Before distributing, the expression has 3 terms: $2(x + 4)$ , $3(x - 1)$ , and $5$ . After distributing: $2x + 8 + 3x - 3 + 5$ , 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 $5x + 10$ , the coefficient of $x$ is 5 and the constant term is 10, so they are equal, but this is coincidental and not generally true for such expressions.
Consider the expression $3x^2 + 7xy - 2x^2 + 5y - 4xy + 8$ . After combining like terms, how many terms in the simplified expression have a coefficient with an absolute value greater than 4?
Answer: 2 terms First, combine like terms: $3x^2 - 2x^2 = x^2$ , $7xy - 4xy = 3xy$ , $5y$ stays the same, and $8$ is the constant. The simplified expression is $x^2 + 3xy + 5y + 8$ . 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.
In the expression $\frac{1}{2}x^3 - 4x^2 + x - \frac{3}{4}$ , which term has the smallest coefficient when all coefficients are written as decimals?
Answer: The $x^2$ term Converting to decimals: $\frac{1}{2} = 0.5$ , $-4 = -4.0$ , $1 = 1.0$ , and $-\frac{3}{4} = -0.75$ . When comparing coefficients as numbers (not absolute values), $-4$ is the smallest since $-4 < -0.75 < 0.5 < 1$ . The $x^2$ term has coefficient $-4$ . Choice A (0.5) and Choice C (1.0) are positive and thus larger. Choice D (-0.75) is negative but larger than -4.
Consider the expression $6x^2y - 4xy + 8x^2y - 12xy + 2$ . After combining like terms, which term has a coefficient that is a factor of all the other coefficients in the simplified expression?
Answer: The constant term 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 $6x^2y - 4xy + 8x^2y - 12xy + 2$ . The $x^2y$ terms are $6x^2y + 8x^2y = 14x^2y$ . The $xy$ terms are $-4xy + (-12xy) = -16xy$ . The constant remains $2$ . So our simplified expression is $14x^2y - 16xy + 2$ . 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 $-16 ÷ 14$ 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 $14 ÷ (-16)$ 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: $14 ÷ 2 = 7$ , $-16 ÷ 2 = -8$ , and $2 ÷ 2 = 1$ . 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.
The expression $-2x^3 + 5x^2 - 3x + 7$ is written in standard form. If this expression is rewritten by factoring out $-x$ from only the terms that contain $x$ , what will be the coefficient of the $x^2$ term in the factored portion?
Answer: 2 Factoring $-x$ from only the terms containing $x$ : $-2x^3 + 5x^2 - 3x = -x(2x^2 - 5x + 3)$ . The complete expression becomes $-x(2x^2 - 5x + 3) + 7$ . In the factored portion $(2x^2 - 5x + 3)$ , the coefficient of $x^2$ 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.
In the expression $4(2x + 3) - 5x + 7(x - 1)$ , which statement correctly identifies the relationship between the terms before any simplification?
Answer: There are exactly 5 terms when written in expanded form, with 3 positive and 2 negative terms Expanding the expression: $4(2x + 3) - 5x + 7(x - 1) = 8x + 12 - 5x + 7x - 7$ . This gives us 5 terms: $+8x$ , $+12$ , $-5x$ , $+7x$ , $-7$ . The positive terms are $8x$ , $12$ , and $7x$ (3 terms), and the negative terms are $-5x$ and $-7$ (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.
In the expression $4x^2 - 6x + 9 - 2x(x + 3)$ , which of the following correctly identifies the terms after expanding but before any combining of like terms?
Answer: Five terms: $4x^2$ , $-6x$ , $9$ , $-2x^2$ , and $-6x$ First, expand $-2x(x + 3) = -2x^2 - 6x$ . The complete expanded expression is $4x^2 - 6x + 9 - 2x^2 - 6x$ , which has 5 terms: $4x^2$ , $-6x$ , $9$ , $-2x^2$ , and $-6x$ . Choice A doesn't expand the last factor. Choice B incorrectly shows $+6x$ instead of $-6x$ from the expansion. Choice D incorrectly separates the factors within the parentheses.
The expression $-3x^2y + 6xy^2 - 9xyz$ can be factored by removing the greatest common factor. What is the coefficient of the $xy^2$ term after factoring out this greatest common factor?
Answer: -2 The greatest common factor is $3xy$ : the GCF of coefficients $|-3|$ , $|6|$ , and $|-9|$ is 3, and the GCF of variables is $xy$ . However, since the first term is negative, we factor out $-3xy$ to avoid a leading negative sign inside the parentheses. Factoring: $-3x^2y + 6xy^2 - 9xyz = -3xy(x - 2y + 3z)$ . The coefficient of $xy^2$ corresponds to the $-2y$ term, so the coefficient is $-2$ . 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.

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