Terms, Factors, & Coefficients - Algebra 1

In the expression $6x(x - 3)$ , what are the factors of $6x(x - 3)$ ?
Answer: $6x\ \text{and}\ (x-3)$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify factors. Factors are parts of an expression that are multiplied together: when you see multiplication (either with · or parentheses next to each other), you're looking at factors. For example, in 4(x + 3), the factors are 4 and (x + 3) because they're being multiplied. In 6x(x - 3), we can see what's being multiplied together: it's 6x times (x - 3). This means the factors are 6x and (x - 3). Remember, factors are connected by multiplication, while terms are connected by addition and subtraction. Choice B is correct because it properly identifies the factors as 6x and (x - 3), following the definition that factors are the multiplied parts. You've got it! Choice A is close, but it confuses terms with factors: it lists 6, x, (x-3) which are smaller pieces, but the question asks for the main factors being multiplied. Remember: terms are added/subtracted, factors are multiplied! Think of terms as 'ingredients being added together' and factors as 'ingredients being multiplied together.' In 6x(x - 3), you're multiplying 6x times (x - 3), so those are factors. But when you expand to 6x² - 18x, you're adding 6x² and -18x, so those are terms. Choice C treats parts inside parentheses as separate terms, but actually (x - 3) is grouped as one factor here. Terms are only separated by + and - at the outermost level!
What is the coefficient of $x^2$ in $4x^3-9x^2+x$ ?
Answer: $-9$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of x² in 4x³ - 9x² + x, we look for the term that contains x². That term is -9x². The coefficient is the number part that's multiplied by the variable, which is -9 with the sign. Choice A is correct because it properly identifies the coefficient as -9, following the definition that coefficients include their sign. You've got it! Choice B is close, but it forgets to include the sign: the coefficient is -9, not 9. The minus sign is part of the coefficient! This is a super common mistake, so watch out for it. To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!
What is the coefficient of $x$ in the expression $x + 5$ ?
Answer: $1$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of x in x + 5, we look for the term that contains x. That term is x (which is like +1x). The coefficient is the number part that's multiplied by the variable, which is 1. If you don't see a number written, like in just 'x', the coefficient is 1! Choice C is correct because it properly identifies the coefficient as 1, following the definition that invisible coefficients are 1 for terms like x. You've got it! Choice D is close, but it gives the variable x when the question asks just for the coefficient (which would be 1, the number part). The coefficient is only the numerical part, not the variable. Choice A misses that when we don't see a number, like in 'x' or '-x', there's still a coefficient—it's 1 for x and -1 for -x. These 'invisible' coefficients are easy to overlook! To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!
How many terms are in the expression $-x^2-4x+1$ ?
Answer: $3$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in 3x² - 5x + 2, there are three terms: 3x², -5x, and 2 (notice that the -5x includes the minus sign). Let's look at the expression -x² - 4x + 1 and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have -x², then -4x, and finally +1. That gives us 3 terms total. Choice C is correct because it properly identifies 3 terms, following the definition that terms are the separate parts connected by + or - signs. You've got it! Choice B counts 4, but let me help clarify: there are only three chunks here; the signs are part of the terms, not extra separators. It's an easy mistake to make when you're learning to identify these parts! Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in 5x² - 3x + 7, circle each + and -, and you can see the three terms clearly!
How many terms are in the expression $5y-3$ ?
Answer: $2$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in 3x² - 5x + 2, there are three terms: 3x², -5x, and 2 (notice that the -5x includes the minus sign). Let's look at the expression 5y - 3 and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have 5y, then -3. That gives us 2 terms total. Choice C is correct because it properly identifies 2 terms, following the definition that terms are the separate parts connected by + or - signs. You've got it! Choice B counts 3, but let me help clarify: there are only two parts here—5y and -3; no extra terms inside. It's an easy mistake to make when you're learning to identify these parts! Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in 5x² - 3x + 7, circle each + and -, and you can see the three terms clearly!
In the expression $5(x - 2)^2 - 3(x - 2) + 7$ , if we let $u = x - 2$ , then the expression becomes $5u^2 - 3u + 7$ . In the original expression, what is the coefficient of the factor $(x - 2)$ that appears with the highest power?
Answer: The coefficient is $5$ because $(x - 2)$ appears squared in the first term. The factor $(x - 2)$ appears to the second power in the term $5(x - 2)^2$ and to the first power in $-3(x - 2)$ . The highest power is 2, and the coefficient of $5(x - 2)^2$ is 5. Choice B identifies the coefficient of the linear factor but not the highest power. Choice C confuses the power with the coefficient. Choice D incorrectly identifies the constant term.
Consider the expression $5x^2 - 3x + 8$ . What are the terms in $5x^2 - 3x + 8$ ?
Answer: $5x^2,\ -3x,\ 8$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in 3x² - 5x + 2, there are three terms: 3x², -5x, and 2 (notice that the -5x includes the minus sign). Let's look at the expression 5x² - 3x + 8 and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have 5x², then -3x (remember to include the sign!), and finally +8. That gives us three terms total. Choice A is correct because it properly identifies the terms as 5x², -3x, 8, following the definition that terms include their signs and are separated by addition or subtraction. You've got it! Choice B is close, but it forgets to include the sign: the term is -3x, not 3x. The minus sign is part of the term! This is a super common mistake, so watch out for it. Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in 5x² - 3x + 8, circle each + and -, and you can see the three terms clearly! Don't forget: signs matter! The term in x² - 4x isn't 4x, it's -4x. Always include the sign that comes right before the term—that sign is part of it. This is one of the most common mistakes in algebra, so being careful here will save you lots of points!
What is the coefficient of $x$ in the expression $x+5$ ?
Answer: $1$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of x in x + 5, we look for the term that contains x. That term is x (or +x). The coefficient is the number part that's multiplied by the variable, which is 1. If you don't see a number written, like in just 'x', the coefficient is 1! Choice C is correct because it properly identifies the coefficient as 1, following the definition that 'invisible' coefficients are 1. You've got it! Choice A gives 0, but that would mean no x term at all; here there is an x with coefficient 1. These 'invisible' coefficients are easy to overlook! To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!
Consider the expression $3xy + 2x - y$ . What are the terms in $3xy + 2x - y$ ?
Answer: $3xy,\ 2x,\ -y$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in 3x² - 5x + 2, there are three terms: 3x², -5x, and 2 (notice that the -5x includes the minus sign). Let's look at the expression 3xy + 2x - y and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have 3xy, then +2x (remember to include the sign!), and finally -y. That gives us three terms total. Choice B is correct because it properly identifies the terms as 3xy, 2x, -y, following the definition that terms include their signs and are separated by addition or subtraction. You've got it! Choice C is close, but it confuses terms with factors: it lists 3, x, y, 2x, -y which are smaller pieces or factors, but the question asks for terms. Remember: terms are added/subtracted, factors are multiplied! Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in 3xy + 2x - y, circle each + and -, and you can see the three terms clearly! Don't forget: signs matter! The term in x² - 4x isn't 4x, it's -4x. Always include the sign that comes right before the term—that sign is part of it. This is one of the most common mistakes in algebra, so being careful here will save you lots of points!
An expression is written as $k \cdot 3^x + k \cdot 3^{x+1} + k \cdot 3^{x+2}$ where $k$ is a constant. If this expression is factored completely, what is the coefficient of $3^x$ in the factored form?
Answer: The coefficient of $3^x$ is $k$ in the factored form $k \cdot 3^x(1 + 3 + 9)$ . First, rewrite using properties of exponents: $k \cdot 3^x + k \cdot 3^{x+1} + k \cdot 3^{x+2} = k \cdot 3^x + k \cdot 3^x \cdot 3 + k \cdot 3^x \cdot 9 = k \cdot 3^x(1 + 3 + 9) = k \cdot 3^x \cdot 13$ . In this factored form $k \cdot 3^x(1 + 3 + 9)$ , the coefficient of $3^x$ is $k$ since we factor out $k \cdot 3^x$ as a unit. Choice A incorrectly identifies $13k$ as the coefficient of $3^x$ rather than the overall coefficient of the entire expression. Choices B and D use incorrect calculations.
How many terms are in the expression $x^2+0x-6$ ?
Answer: $3$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in $3x^2 - 5x + 2$ , there are three terms: $3x^2$ , $-5x$ , and $2$ (notice that the $-5x$ includes the minus sign). Let's look at the expression $x^2 + 0x - 6$ and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have $x^2$ , then $+0x$ (remember to include the sign!), and finally $-6$ . That gives us 3 terms total. Choice C is correct because it properly identifies 3 terms, following the definition that terms are the chunks connected by + or -. You've got it! Choice A counts only 2, but let me help clarify: even though $0x$ is zero, it's still written as a separate term here. It's an easy mistake to make when you're learning to identify these parts! Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in $x^2 + 0x - 6$ , circle each + and -, and you can see the three terms clearly!
What is the coefficient of $y$ in the expression $2y^2-y+6$ ?
Answer: $-1$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of y in 2y² - y + 6, we look for the term that contains y (without higher powers). That term is -y. The coefficient is the number part that's multiplied by the variable, which is -1. If you don't see a number written, like in just 'x', the coefficient is 1! Choice B is correct because it properly identifies the coefficient as -1, following the definition that 'invisible' coefficients include the sign. You've got it! Choice C misses that when we don't see a number, like in '-y', there's still a coefficient—it's -1. These 'invisible' coefficients are easy to overlook! To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!
What is the coefficient of $x$ in the expression $7x^2-4x+1$ ?
Answer: $-4$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of x in 7x² - 4x + 1, we look for the term that contains x. That term is -4x. The coefficient is the number part that's multiplied by the variable, which is -4. If you don't see a number written, like in just 'x', the coefficient is 1! Choice A is correct because it properly identifies the coefficient as -4, following the definition that coefficients include their sign. You've got it! Choice B is close, but it forgets to include the sign: the term is -4x, not 4x. The minus sign is part of the coefficient! This is a super common mistake, so watch out for it. To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!
In the expression $-8n^2+3n+10$ , what is the coefficient of $n^2$ ?
Answer: $-8$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of n² in -8n² + 3n + 10, we look for the term that contains n². That term is -8n². The coefficient is the number part that's multiplied by the variable, which is -8 with the sign. Choice B is correct because it properly identifies the coefficient as -8, following the definition that coefficients include their sign. You've got it! Choice A is close, but it forgets to include the sign: the coefficient is -8, not 8. The minus sign is part of the coefficient! This is a super common mistake, so watch out for it. To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like n) or -1 (for terms like -n). Write out that 'invisible 1' when learning, and it'll help!
In the expression $9t^2-3t-4$ , what is the coefficient of the quadratic term?
Answer: $9$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of the quadratic term in 9t² - 3t - 4, we look for the term that contains t². That term is 9t². The coefficient is the number part that's multiplied by the variable, which is 9. Choice B is correct because it properly identifies the coefficient as 9, following the definition that it's for the quadratic (t²) term. You've got it! Choice A gives the coefficient of t, which is -3, but the question asks for the quadratic one. It's an easy mistake to make when you're learning to identify these parts! To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like t) or -1 (for terms like -t). Write out that 'invisible 1' when learning, and it'll help!
The cost (in dollars) to rent a bike is given by $C=3h+8$ , where $h$ is the number of hours. What is the constant term in $3h+8$ ?
Answer: $8$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify the constant term. The constant term is the part of the expression that doesn't have any variables—it's just a number by itself, like the +2 at the end of 3x² - 5x + 2. The constant term is the number that stands alone without any variables attached. In 3h + 8, looking through each part, the constant term is +8. This is different from the coefficients, which are the numbers multiplied by variables. Choice C is correct because it properly identifies the constant term as 8, following the definition that it's the number without variables. You've got it! Choice A forgets that the constant term is the number without any variables. In this expression, it's 8, not 3 (which is the coefficient of h). A quick check: count your terms by counting how many parts would be separated if you put the expression in a calculator with clear + and - between them. For 3h + 8, you'd enter it as two separate chunks connected by operations—those are your two terms, and the constant is the one without h!
What are the terms in $4a^2 - a + 7$ ?
Answer: $4a^2,\;-a,\;7$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in $3x^2 - 5x + 2$ , there are three terms: $3x^2$ , $-5x$ , and $2$ (notice that the $-5x$ includes the minus sign). Let's look at the expression $4a^2 - a + 7$ and find the terms. Terms are separated by + or - signs (at the main level, not inside parentheses): starting from the left, we have $4a^2$ , then $-a$ , and finally $+7$ . That gives us 3 terms total. Choice A is correct because it properly identifies the terms as $4a^2$ , $-a$ , $7$ , following the definition that terms include their signs. You've got it! Choice B is close, but it forgets to include the sign: the term is $-a$ , not a. The minus sign is part of the term! This is a super common mistake, so watch out for it. Don't forget: signs matter! The term in $x^2 - 4x$ isn't 4x, it's $-4x$ . Always include the sign that comes right before the term—that sign is part of it. This is one of the most common mistakes in algebra, so being careful here will save you lots of points!
The area of a rectangle is modeled by $A=2w^2+5w$ , where $w$ is the width. What is the coefficient of $w$ ?
Answer: $5$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of w in 2w² + 5w, we look for the term that contains w (not w²). That term is +5w. The coefficient is the number part that's multiplied by the variable, which is 5. Choice B is correct because it properly identifies the coefficient as 5, following the definition that it's the number multiplied by w. You've got it! Choice A gives the coefficient of w² instead, which is 2, but the question asks for the coefficient of w. It's an easy mistake to make when you're learning to identify these parts! To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like w) or -1 (for terms like -w). Write out that 'invisible 1' when learning, and it'll help!
In the expression $-3(x-2)(x+1)$ , what is being multiplied?
Answer: $-3$ , $(x-2)$ , and $(x+1)$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify factors. Factors are parts of an expression that are multiplied together: when you see multiplication (either with · or parentheses next to each other), you're looking at factors. For example, in 4(x + 3), the factors are 4 and (x + 3) because they're being multiplied. In -3(x - 2)(x + 1), we can see what's being multiplied together: -3 times (x - 2) times (x + 1). This means the factors are -3, (x - 2), and (x + 1). Remember, factors are connected by multiplication, while terms are connected by addition and subtraction. Choice A is correct because it properly identifies the factors as -3, (x - 2), and (x + 1), following the definition that factors are multiplied. You've got it! Choice C treats parts inside parentheses as separate terms, but actually (x - 2) and (x + 1) are grouped as factors here. Terms are only separated by + and - at the outermost level! Think of terms as 'ingredients being added together' and factors as 'ingredients being multiplied together.' In -3(x - 2)(x + 1), you're multiplying -3, (x - 2), and (x + 1), so those are factors. But if you expand, you'd get terms being added.
In the expression $-x^2 + 3x - 4$ , what is the leading coefficient?
Answer: $-1$ This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the leading coefficient in -x² + 3x - 4, we look for the term with the highest power, which is -x² (like -1x²). That term is -x². The coefficient is the number part that's multiplied by the variable, which is -1 (with sign). If you don't see a number written, like in just 'x', the coefficient is 1! Choice B is correct because it properly identifies the leading coefficient as -1, following the definition that invisible coefficients with signs are -1 for terms like -x². You've got it! Choice D is close, but it forgets to include the sign and misses the leading term: the leading coefficient is -1 for -x², not -4 (which is the constant). This is a super common mistake, so watch out for it. Choice C gives the entire term -x² when the question asks just for the coefficient (which would be -1, the number part). The coefficient is only the numerical part, not the variable. To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like x) or -1 (for terms like -x). Write out that 'invisible 1' when learning, and it'll help!

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