Terms, Factors, & Coefficients - Algebra 2

1. In the expression $3(2x+1)^2$, what is the coefficient of $(2x+1)^2$?​
   Answer: $3$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). In the expression 3(2x+1)², we need to find the coefficient of (2x+1)². The expression shows 3 multiplied by (2x+1)², so 3 is the numerical factor that multiplies the entire quantity (2x+1)². The coefficient is 3! Choice B correctly identifies 3 as the coefficient, recognizing that it's the number multiplying the squared binomial. Choice A (1) might come from thinking there's no visible coefficient, but we clearly see the 3. Choice C or D might confuse the base expression (2x+1) with the coefficient, but the coefficient is the numerical multiplier of the entire term. When a number multiplies a more complex expression, that number is still the coefficient!
2. What is the coefficient of $x^2$ in the polynomial $7x^3 - 5x^2 + 2x - 1$?
   Answer: $-5$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify coefficients, which are the numerical multipliers of variable parts in a term. Coefficients are the numerical factors in each term: in 7x³ - 5x² + 2x - 1, the coefficient of x³ is 7, of x² is -5, of x is 2, and the constant term is -1 (which is like -1 times x⁰). Remember, the sign is part of the coefficient, so -5x² means the coefficient is -5, not just 5 with a minus sign separate. Terms are separated by + or -, but here we're honing in on the coefficient of a specific power, x². In this polynomial, the x² term is -5x², so its coefficient is -5—easy to spot once you include the sign! Choice A correctly identifies the coefficient as -5 by recognizing the negative sign as part of it. Choice B is a tempting distractor because it ignores the sign and picks the absolute value 5, but remember, coefficients include signs—think of it as the number you'd pull out when factoring! To find coefficients quickly, rewrite the expression with all signs attached to the numbers: 7x³ + (-5)x² + 2x + (-1), and you'll never miss one. You're doing awesome at this—keep building that polynomial intuition!
3. In the expression $4x^2 + 3x$, is the number $4$ best described as a term, factor, or coefficient?
   Answer: Coefficient This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). In the expression 4x² + 3x, we need to classify what the number 4 represents. Looking at the first term 4x², we see that 4 is multiplied by x², making 4 the numerical multiplier of the variable part x². This means 4 is the coefficient of x² in the first term! Choice C correctly identifies 4 as a coefficient, recognizing its role as the numerical multiplier of x². Choice A (term) would be incorrect because 4x² is the complete term, not just 4. Choice B (factor) is partially correct since 4 is a factor within the term 4x², but "coefficient" is the more specific and accurate description. When a number multiplies a variable expression, it's best described as a coefficient!
4. In the expression $5(x - 1)^2 + 2x(x + 3) - 4$, identify all the terms. (Do not expand; terms are separated by $+$ or $-$ at the top level.)
   Answer: Terms: $5(x - 1)^2$, $2x(x + 3)$, $-4$ This question tests your understanding of identifying terms without expanding—specifically, recognizing top-level additive parts. Terms are separated by + or - signs at the outermost level; for example, in 2(a + b) - 3, there are two terms: 2(a + b) and -3, treating parenthetical groups as single units. In 5(x - 1)² + 2x(x + 3) - 4, the terms are 5(x - 1)², +2x(x + 3), and -4, as the + and - separate them without going inside parentheses. Choice A correctly lists these three terms by respecting the top-level separation and not expanding. A tempting distractor like Choice B breaks down the coefficients and parenthetical parts as separate terms, but that's confusing factors with terms—parentheses group factors within a term. A good strategy is to ignore what's inside parentheses and count only the outer + and - signs: here, two separating signs (+, -) mean 2 + 1 = 3 terms. You're building a strong foundation here—keep practicing to master expression structure!
5. In the expression $5(x-1)^2 + 2x(x+3) - 4$, identify all the terms (additive parts separated by $+$ or $-$).​
   Answer: $5(x-1)^2,\ 2x(x+3),\ -4$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). Terms are the pieces of an expression separated by plus or minus signs: in the expression 5(x-1)² + 2x(x+3) - 4, we need to identify what's being added or subtracted. Looking at the expression, we see three operations at the "top level": first we have 5(x-1)², then we add 2x(x+3), then we subtract 4. This gives us three terms: 5(x-1)², 2x(x+3), and -4 (the minus sign belongs to the 4). Choice A correctly identifies these three terms by listing each complete multiplicative unit that's being added or subtracted. Choice B incorrectly breaks apart the factors within each term—it's listing parts that are multiplied together, not the terms themselves. Remember: terms are separated by + or -, while factors are connected by multiplication!
6. In the expression $2xy^2 - 3x^2y + 5y$, what is the coefficient of $xy^2$?
   Answer: $2$ This question tests your understanding of the structure of algebraic expressions—specifically, identifying the coefficient of a specific variable term like xy² in 2xy² - 3x²y + 5y. Coefficients are the numerical multipliers: the term 2xy² has coefficient 2 (multiplying xy²), -3x²y has -3 (for x²y), and 5y has 5 (for y)—we pick the one matching xy². Terms are the additive parts: 2xy², -3x²y, +5y, each with its coefficient including the sign. Factors within a term would be like 2, x, y² for the first, but we're after the coefficient of a particular monomial. Remember, even if variables are similar, we match exactly—xy² is different from x²y! Choice A correctly identifies 2 as the coefficient by spotting the matching term 2xy². Choice B is a tempting distractor because it includes the y², but that's part of the variable, not the coefficient—coefficients are just the numbers! To find a specific coefficient, scan for the exact variable combo and grab its numerical multiplier, including sign; for example, in ax² + bxy + cy, coefficient of xy is b. You're nailing this—keep up the great effort!
7. What are the factors of the term $6x^2y$? (Factors are parts multiplied together.)
   Answer: $6,\ x^2,\ y$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). Factors are parts multiplied together within a term: in the term 6x²y, we need to identify all the parts being multiplied. We can write 6x²y as 6 × x² × y, or even more explicitly as 6 × x × x × y. The factors are 6, x², and y (or we could list x twice instead of x²). Choice A correctly identifies the factors as 6, x², and y, recognizing that x² is a valid way to express the factor (though 6, x, x, y would also be correct). Choice D lists 6, x, y but misses that there are two x's being multiplied (x² means x × x). When identifying factors, remember to account for all parts being multiplied, including repeated factors shown as exponents!
8. In the expression $\frac{6x^3 - 9x^2 + 12x}{3x}$, after simplification, how many terms have a coefficient that is a prime number?
   Answer: Exactly one term has a prime coefficient Simplify the expression: $\frac{6x^3 - 9x^2 + 12x}{3x} = \frac{6x^3}{3x} - \frac{9x^2}{3x} + \frac{12x}{3x} = 2x^2 - 3x + 4$. The coefficients are $2$, $-3$, and $4$. Only $2$ is prime ($-3$ is negative and $4 = 2^2$ is composite). Choice B might count $|-3| = 3$ as prime. Choice C incorrectly considers all coefficients prime. Choice D overlooks that $2$ is prime.
9. In the rational expression $\dfrac{3x^2 - 2x + 5}{x-1}$, identify the terms in the numerator.
   Answer: $3x^2,\ -2x,\ 5$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). In the rational expression (3x² - 2x + 5)/(x-1), we need to identify the terms specifically in the numerator. The numerator is 3x² - 2x + 5, and terms are separated by + or - signs. We have: first term 3x², second term -2x (the minus sign belongs to this term), and third term 5 (or +5). That's three terms in the numerator! Choice A correctly lists 3x², -2x, and 5 as the three terms, properly including the negative sign with the middle term. Choice B incorrectly combines the first two terms, while Choice D breaks the terms into their factors. Remember: we're looking for terms (additive parts) in just the numerator, not factors or anything involving the denominator!
10. In the expression $3(2x + 1)^2$, what is the coefficient of $(2x + 1)^2$?
    Answer: $3$ This question tests your understanding of coefficients when a parenthetical expression is treated as a variable part. Coefficients are the numerical multipliers in front of variable or grouped parts; for instance, in 4(y + 2), the coefficient of (y + 2) is 4. In the expression 3(2x + 1)², the entire (2x + 1)² is being multiplied by 3, so the coefficient is 3—think of (2x + 1)² as a single unit like a variable. Choice B correctly identifies 3 as the coefficient by recognizing it as the numerical factor. A tempting distractor like Choice A might confuse the coefficient with the inside of the parentheses, but remember, the coefficient is outside, multiplying the whole group. To spot this, rewrite as 3 × (2x + 1)², highlighting the coefficient clearly. This concept is useful for factoring and expanding—great job tackling it, and keep going!
11. Distinguish terms (additive parts) from factors (multiplicative parts): In the expression $2x(x - 3)^2 + 5$, which statement is correct?
    Answer: There are 2 terms: $2x(x - 3)^2$ and $5$. This question tests your ability to distinguish terms from factors in an expression with multiplication and addition. Terms are additive parts separated by + or -, while factors are multiplied within terms; for example, in 3a(b + 1) + 2, there are two terms: 3a(b + 1) and +2. In 2x(x - 3)² + 5, the terms are 2x(x - 3)² and +5, as the + separates them, with 2x and (x - 3)² being factors in the first term. Choice B correctly states there are 2 terms by identifying the top-level addition. A tempting distractor like Choice A might split the factors as separate terms, but parentheses and multiplication keep them together—count outer + and - only. Strategy: cover parentheses and see the structure: it's like A + B, where A is 2x(something) and B is 5. This distinction is crucial for algebra— you're doing an amazing job!
12. Consider the expression $\sqrt{x}(4\sqrt{x} - 3) + 2x - 7$. When this expression is written in terms of integer powers of $x$, how many distinct terms are there?
    Answer: There are three distinct terms in the expression Expand and simplify: $\sqrt{x}(4\sqrt{x} - 3) + 2x - 7 = 4x - 3\sqrt{x} + 2x - 7 = 6x - 3\sqrt{x} - 7$. Since $\sqrt{x} = x^{1/2}$, we can write this as $6x^1 - 3x^{1/2} - 7x^0$. There are three distinct terms with different powers of $x$: $6x$, $-3\sqrt{x}$, and $-7$. Choice A counts terms before combining like terms. Choice B undercounts by combining unlike terms. Choice C overcounts by not recognizing like terms.
13. In the expression $2x^3 - 5x^2 + 3x - 1$, suppose the coefficient of the $x^2$ term is changed to its opposite. What would be the new coefficient of $x^2$, and how would this affect the sum of all coefficients?
    Answer: The new $x^2$ coefficient is $5$, and the sum increases by $10$ The original coefficient of $x^2$ is $-5$, so its opposite is $5$. The original sum of coefficients is $2 + (-5) + 3 + (-1) = -1$. The new sum would be $2 + 5 + 3 + (-1) = 9$. The increase is $9 - (-1) = 10$. Choice B incorrectly calculates the change as just the difference between $5$ and $0$. Choice C doesn't change the sign. Choice D gives the wrong direction of change.
14. In the expression $5(x-1)^2 + 2x(x+3) - 4$, which list correctly identifies all the terms? (Do not expand.)
    Answer: Terms: $5(x-1)^2$, $2x(x+3)$, $-4$ This question tests your understanding of the structure of algebraic expressions—specifically, identifying terms without expanding, treating multiplied groups as single terms. Terms are separated by + or -: in 5(x-1)² + 2x(x+3) - 4, they are 5(x-1)², +2x(x+3), and -4—three terms, keeping parentheses intact. Signs belong to terms, so positive first, positive second, negative third. Factors are within terms, like 5 and (x-1)² in the first, but the question asks for terms, not factors or coefficients. Don't break squares or products into more terms unless there's + or - inside! Choice B correctly identifies the three terms by grouping the multiplied parts correctly. Choice A is a tempting distractor because it lists factors like 5 and (x-1)² as separate terms, but multiplication doesn't create new terms—only addition/subtraction does. Strategy: count + and - signs (two here: + and -), add 1 for the first term, getting three; list each cluster between them. You're doing brilliantly—keep shining!
15. What is the coefficient of $x^2$ in the polynomial $3x^3 - 7x^2 + 2x - 9$?
    Answer: $-7$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). The coefficient is the numerical factor that multiplies the variable part: in the polynomial 3x³ - 7x² + 2x - 9, we need to find what number multiplies x². Looking at the second term, we have -7x², which means -7 times x². The coefficient of x² is therefore -7, including the negative sign! Choice B correctly identifies -7 as the coefficient, recognizing that the sign is part of the coefficient. Choice A (7) forgets to include the negative sign—remember, coefficients include their signs! Choice C (-7x) includes the variable, but a coefficient is just the numerical part. When identifying coefficients, always include the sign but exclude the variable part!
16. Consider the expression $3x^2(2x - 5) + 7x - 4$. When this expression is written in standard form, how many terms contain $x$ as a factor?
    Answer: Three terms contain $x$ as a factor First, expand the expression: $3x^2(2x - 5) + 7x - 4 = 6x^3 - 15x^2 + 7x - 4$. In standard form, the terms are $6x^3$, $-15x^2$, $7x$, and $-4$. Three of these terms ($6x^3$, $-15x^2$, and $7x$) contain $x$ as a factor. Choice A counts only two terms, missing one of the variable terms. Choice C incorrectly includes the constant term. Choice D significantly undercounts the variable terms.
17. In the expression $2x(x-3)^2 + 5$, which of the following correctly distinguishes the terms and the factors of the first term? First term: $2x(x-3)^2$
    Answer: Terms: $2x(x-3)^2$ and $5$; Factors of first term: $2$, $x$, and $(x-3)^2$ This question tests your understanding of the structure of algebraic expressions—specifically, distinguishing terms (additive parts) from factors (multiplicative parts within a term). Terms are separated by + or -, so in 2x(x-3)² + 5, there are two terms: 2x(x-3)² and +5, treating the parenthetical as a unit. Factors within the first term are 2, x, and (x-3)², since they're multiplied together—note that (x-3)² is one factor, even though it's (x-3)*(x-3) inside. Coefficients are numerical multipliers, like 2 here, but we're identifying terms and factors. Remember, don't expand unless told to; keep groups intact for term counting! Choice B correctly identifies the two terms and the factors of the first term by keeping the multiplied parts together but listing them separately as factors. Choice A is a tempting distractor because it treats 2x and (x-3)² as separate terms, but there's no + or - between them—it's multiplication, so one term! For transferable strategy, first count terms by + and - separators (here, one + , so two terms), then for factors in a term, list what's multiplied: like in a b (c+d), factors are a, b, (c+d). You're doing fantastic—keep practicing, and it'll become second nature!
18. The expression $3x^2y + xy^2 - 6x^2y + 2xy^2$ can be simplified by combining like terms. After simplification, what is the product of all the coefficients?
    Answer: The product of all coefficients is $-9$ When you encounter an expression with multiple terms, your first step is to identify and combine like terms—terms that have identical variable parts with the same exponents. Let's work through this systematically. The expression is $3x^2y + xy^2 - 6x^2y + 2xy^2$. First, identify the like terms by looking at the variable parts: $3x^2y$ and $-6x^2y$ are like terms (both have $x^2y$), while $xy^2$ and $2xy^2$ are like terms (both have $xy^2$). Now combine the coefficients of like terms. For the $x^2y$ terms: $3x^2y - 6x^2y = -3x^2y$. For the $xy^2$ terms: $xy^2 + 2xy^2 = 3xy^2$. The simplified expression is $-3x^2y + 3xy^2$. The coefficients are $-3$ and $3$, so their product is $(-3) \times 3 = -9$. This confirms answer C is correct. Let's examine why the other answers are wrong. Answer A gives $9$, which would result from multiplying the absolute values $3 \times 3$ but ignoring the negative sign. Answer B gives $6$, which might come from incorrectly adding the coefficients $(-3) + 3 + 6 = 6$ instead of multiplying the final coefficients. Answer D gives $-6$, which could result from various arithmetic errors in the combining process. Remember: always combine like terms first, then perform the requested operation on the final simplified form. Don't work with the original coefficients directly.
19. List all terms and their coefficients in the expression $-x^2 + \frac{2}{3}x - 5$. (Remember $-x^2$ has an implicit coefficient of $-1$.)
    Answer: Terms: $-x^2,\ \frac{2}{3}x,\ -5$; Coefficients: $-1,\ \frac{2}{3},\ -5$ This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). In -x² + (2/3)x - 5, we need to list all terms and their coefficients. The terms are: -x² (first term), (2/3)x (second term), and -5 (third term). For coefficients: -x² has an implicit coefficient of -1 (since -x² means -1 × x²), (2/3)x has coefficient 2/3, and the constant term -5 can be thought of as having coefficient -5. Choice A correctly identifies the three terms as -x², (2/3)x, and -5, with coefficients -1, 2/3, and -5 respectively. Choice B incorrectly drops the negative signs from the terms and coefficients. Remember: when you see -x² with no visible number, the coefficient is -1, not 1! Always include signs when listing terms and coefficients!
20. How many terms are in the expression $2x^3 - 5x^2 + x - 8$? (Terms are separated by $+$ or $-$.)​
    Answer: 4 terms This question tests your understanding of the structure of algebraic expressions—specifically, how to identify terms (parts separated by addition or subtraction), factors (parts connected by multiplication), and coefficients (numerical multipliers of variable parts). Terms are the pieces of an expression separated by plus or minus signs: in 2x³ - 5x² + x - 8, we need to count how many separate pieces there are. Looking at the expression, we can identify: first term is 2x³, second term is -5x² (the minus sign belongs to this term), third term is x (or +x), and fourth term is -8. That's 4 terms total! Choice B correctly identifies 4 terms by counting each complete piece between the addition and subtraction operations. Choice A (3 terms) might come from missing one term or incorrectly combining terms, while Choice C (5 terms) might come from counting factors instead of terms. Here's a quick counting method: count the + and - signs that separate terms (we have 3: -, +, -), then add 1 to get the total number of terms: 3 + 1 = 4 terms!

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