Understanding and Operating with Polynomials - Algebra 1

1. Multiply: $(x + 6)(x - 3)$
   Answer: $x^2 + 3x - 18$ This question tests your understanding of how to multiply polynomials using the key ideas of distributing and using FOIL. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. For binomials, we often use FOIL (First, Outer, Inner, Last) to remember all four products we need to find, then we combine any like terms in the result. Multiplying (x + 6)(x - 3), we use FOIL: First: x · x = x², Outer: x · (-3) = -3x, Inner: 6 · x = 6x, Last: 6 · (-3) = -18. This gives us x² - 3x + 6x - 18. Now combine like terms: -3x + 6x = 3x, so we get x² + 3x - 18. Each term from the first binomial gets multiplied by each term from the second! Choice A is correct because it multiplies each term correctly and combines like terms properly, giving us x² + 3x - 18 in standard form. Excellent! Choice C has a sign error: when we combine -3x + 6x, we get +3x, not -3x. When combining like terms with different signs, we subtract the smaller coefficient from the larger and keep the sign of the larger coefficient! For multiplication, especially with binomials, FOIL is your friend: First (first terms of each), Outer (outer terms), Inner (inner terms), Last (last terms). Then don't forget to combine any like terms at the end—many students get all four products right but forget this final step!
2. What is the sum of $(3x^3 - 2x^2 + x - 5)$ and $(x^3 + 4x^2 - 3x + 2)$?
   Answer: $4x^3 + 2x^2 - 2x - 3$ This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x³ and x³ are like terms (both have x³), so they combine to 4x³, but x³ and x² are NOT like terms because the exponents are different. Let's add (3x³ - 2x² + x - 5) + (x³ + 4x² - 3x + 2) by lining up like terms: the x³ terms are 3x³ and x³, which combine to 4x³. The x² terms are -2x² and 4x², which combine to 2x². The x terms are x and -3x, which combine to -2x. The constant terms are -5 and 2, which combine to -3. Putting it all together: 4x³ + 2x² - 2x - 3. Choice A is correct because it combines all like terms properly, giving us 4x³ + 2x² - 2x - 3 in standard form. Excellent! Choice B makes an error with the constant terms: it calculates -5 + 2 = 7, but -5 + 2 = -3. Remember that when adding a negative and a positive number, we find the difference and keep the sign of the larger absolute value! The key to polynomial addition and subtraction: line up your like terms in columns (all the x³ terms together, all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
3. Add: $(2x^2 - 3x + 8) + (-x^2 + 6x - 5)$
   Answer: $x^2 + 3x + 3$ This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (2x² - 3x + 8) + (-x² + 6x - 5) by lining up like terms: the x² terms are 2x² and -x², which combine to x². The x terms are -3x and 6x, which combine to 3x. The constant terms are 8 and -5, which combine to 3. Putting it all together: x² + 3x + 3. Choice A is correct because it combines all like terms properly, giving us x² + 3x + 3 in standard form. Excellent! Choice B makes an error when combining the x terms: -3x + 6x = 3x, not 9x. When combining -3x and 6x, think of it as 6x - 3x = 3x, not adding their absolute values! The key to polynomial addition: line up your like terms in columns (all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
4. Find the product: $(x^2 + 2x - 3)(x + 4)$
   Answer: $x^3 + 6x^2 + 5x - 12$ This question tests your understanding of how to multiply polynomials using the key ideas of distributing each term. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. For binomials, we often use FOIL (First, Outer, Inner, Last) to remember all four products we need to find, then we combine any like terms in the result. Multiplying (x² + 2x - 3)(x + 4), we distribute: x²(x + 4) = x³ + 4x², 2x(x + 4) = 2x² + 8x, -3(x + 4) = -3x - 12. Now combine like terms: x³ + (4x² + 2x²) + (8x - 3x) - 12 = x³ + 6x² + 5x - 12. Each term from the first polynomial gets multiplied by each term from the second! Choice A is correct because it multiplies each term correctly and combines the like terms, giving us x³ + 6x² + 5x - 12 in standard form. Excellent! Choice B has a sign error: when multiplying -3 by x, we get -3x, but combined with +8x it's +5x, not -5x. Multiplying two negatives gives a positive, and a negative times a positive gives a negative—those sign rules matter here! To check your polynomial multiplication, count the terms before combining: here, distributing a trinomial to a binomial should give six products initially, then combine like terms. If you don't get the right number of initial products, you missed one!
5. Add: $\left(3x^2 - 5x + 7\right) + \left(2x^2 + 4x - 9\right)$. Give your answer in simplest form.
   Answer: $5x^2 - x - 2$ This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (3x² - 5x + 7) + (2x² + 4x - 9) by lining up like terms: the x² terms are 3x² and 2x², which combine to 5x². The x terms are -5x and 4x, which combine to -x. The constant terms are 7 and -9, which combine to -2. Putting it all together: 5x² - x - 2. Choice A is correct because it combines all like terms properly, giving us 5x² - x - 2 in standard form. Excellent! Choice B makes a calculation error with the x terms, getting -9x instead of -x. When combining -5x + 4x, we get -1x or just -x, not -9x—remember to add the coefficients carefully! The key to polynomial addition: line up your like terms in columns (all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
6. If $R(x) = x^3 - 2x + 5$ and $S(x) = 3x^2 + x - 4$, which expression represents $R(x) - S(x)$?
   Answer: $x^3 - 3x^2 - 3x + 9$ To find $R(x) - S(x)$, we subtract each term of $S(x)$ from the corresponding terms in $R(x)$: $R(x) - S(x) = (x^3 - 2x + 5) - (3x^2 + x - 4) = x^3 - 2x + 5 - 3x^2 - x + 4 = x^3 - 3x^2 + (-2x - x) + (5 + 4) = x^3 - 3x^2 - 3x + 9$. Choice B incorrectly adds $3x^2$ instead of subtracting it. Choice C makes an error in combining the $x$ terms, getting $-x$ instead of $-3x$. Choice D has errors in both the $x$ terms ($+3x$ instead of $-3x$) and the constant term ($+1$ instead of $+9$).
7. What is the sum of $2x^3 - x^2 + 4x - 1$ and $-x^3 + 3x^2 - 6x + 5$?
   Answer: $x^3 + 2x^2 - 2x + 4$ This question tests your understanding of how to add polynomials using the key ideas of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, $3x^2$ and $5x^2$ are like terms (both have $x^2$), so they combine to $8x^2$, but $3x^2$ and $3x$ are NOT like terms because the exponents are different. Let's add $2x^3 - x^2 + 4x - 1$ and $-x^3 + 3x^2 - 6x + 5$ by lining up like terms: the $x^3$ terms are $2x^3$ and $-x^3$, which combine to $x^3$; the $x^2$ terms are $-x^2$ and $3x^2$ to $2x^2$; the $x$ terms are $4x$ and $-6x$ to $-2x$; the constants are $-1$ and $5$ to $4$. Putting it all together: $x^3 + 2x^2 - 2x + 4$. Choice A is correct because it combines all like terms properly, giving us $x^3 + 2x^2 - 2x + 4$ in standard form. Excellent! Choice C makes a really common mistake: it tries to combine unlike terms or mishandles signs, like turning $-x^2 + 3x^2$ into $-4x^2$ instead of $2x^2$. Remember, you can only combine terms that have exactly the same variable part—$3x^2$ and $5x^2$ can combine, but $x^2$ and $x$ cannot! The key to polynomial addition and subtraction: line up your like terms in columns (all the $x^3$ terms together, all the $x^2$ terms together, all the $x$ terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
8. Multiply: $(x + 5)(x - 3)$
   Answer: $x^2 + 2x - 15$ This question tests your understanding of how to multiply polynomials using the key ideas of distributing and using FOIL. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. For binomials, we often use FOIL (First, Outer, Inner, Last) to remember all four products we need to find, then we combine any like terms in the result. Multiplying (x + 5)(x - 3), we use FOIL: First x·x = x², Outer x·(-3) = -3x, Inner 5·x = 5x, Last 5·(-3) = -15. Now combine like terms: x² + (-3x + 5x) - 15 = x² + 2x - 15. Each term from the first binomial gets multiplied by each term from the second! Choice A is correct because it multiplies each term correctly and combines the like terms, giving us x² + 2x - 15 in standard form. Excellent! Choice B misses one of the products in the multiplication—an easy thing to do! When multiplying binomials, we need all four products: First, Outer, Inner, and Last. This choice has -2x instead of +2x, perhaps forgetting to add the outer and inner properly. For multiplication, especially with binomials, FOIL is your friend: First (first terms of each), Outer (outer terms), Inner (inner terms), Last (last terms). Then don't forget to combine any like terms at the end—many students get all four products right but forget this final step!
9. Multiply: $\left(x + 5\right)\left(x - 3\right)$
   Answer: $x^2 + 2x - 15$ This question tests your understanding of how to multiply polynomials using the key idea of FOIL. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. For binomials, we often use FOIL (First, Outer, Inner, Last) to remember all four products we need to find, then we combine any like terms in the result. Multiplying (x + 5)(x - 3), we use FOIL: First: x · x = x², Outer: x · (-3) = -3x, Inner: 5 · x = 5x, Last: 5 · (-3) = -15. This gives us x² - 3x + 5x - 15. Now combine like terms: -3x + 5x = 2x, so we get x² + 2x - 15. Each term from the first binomial gets multiplied by each term from the second! Choice A is correct because it multiplies each term correctly and combines like terms properly, giving us x² + 2x - 15 in standard form. Excellent! Choice B has a sign error when combining the middle terms: -3x + 5x = 2x, not -2x. When combining like terms with different signs, we subtract the smaller coefficient from the larger and keep the sign of the larger—here 5x is larger than 3x, so we get positive 2x! For multiplication, especially with binomials, FOIL is your friend: First (first terms of each), Outer (outer terms), Inner (inner terms), Last (last terms). Then don't forget to combine any like terms at the end—many students get all four products right but forget this final step!
10. Find the product: $(x^2 - x + 3)(x + 2)$
    Answer: $x^3 + x^2 + x + 6$ This question tests your understanding of how to multiply a trinomial by a binomial using the distributive property. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. When multiplying (x² - x + 3)(x + 2), we distribute each term from the first polynomial to both terms in the second: x² · x = x³, x² · 2 = 2x², (-x) · x = -x², (-x) · 2 = -2x, 3 · x = 3x, 3 · 2 = 6. This gives us x³ + 2x² - x² - 2x + 3x + 6. Now combine like terms: 2x² - x² = x² and -2x + 3x = x, so we get x³ + x² + x + 6. Choice A is correct because it multiplies each term correctly and combines all like terms properly, giving us x³ + x² + x + 6 in standard form. Excellent! Choice B makes an error when combining the x terms: -2x + 3x = x, not 5x. When combining like terms with different signs, we need to be extra careful with our arithmetic! To check your polynomial multiplication, count the terms before combining: a trinomial times a binomial should give you 6 products initially (3 × 2 = 6), and then you combine any like terms to get your final answer. If you don't get the right number of initial products, you missed one!
11. Consider the polynomial expression $(x^2 + kx + 3)(2x - 1)$ where $k$ is a constant. If the coefficient of the $x^2$ term in the expanded form is $7$, what is the value of $k$?
    Answer: $k = 4$ When you see a polynomial multiplication problem asking for a specific coefficient, you need to expand strategically and focus only on the terms that will contribute to that coefficient. To find the coefficient of the $x^2$ term in $(x^2 + kx + 3)(2x - 1)$, you only need to identify which multiplication combinations will produce $x^2$ terms. Using the distributive property (FOIL), there are two ways to get $x^2$: - $x^2 \cdot (-1) = -x^2$ - $kx \cdot 2x = 2kx^2$ The total coefficient of $x^2$ is $-1 + 2k$. Since this coefficient equals $7$: $-1 + 2k = 7$ $2k = 8$ $k = 4$ Let's check why the other answers are wrong: A) If $k = 8$, then $2k - 1 = 16 - 1 = 15$, not $7$. B) If $k = 5$, then $2k - 1 = 10 - 1 = 9$, not $7$. C) If $k = 6$, then $2k - 1 = 12 - 1 = 11$, not $7$. These wrong answers likely come from common algebraic mistakes: forgetting the negative coefficient from $x^2 \cdot (-1)$, or miscalculating the distributive property. Study tip: When finding specific coefficients in polynomial multiplication, don't expand everything unnecessarily. Identify only the term combinations that produce the degree you need, then set up your equation. This saves time and reduces calculation errors on exams.
12. Which expression is equivalent to $(x^2 - 3x + 2) + (2x^2 + x - 6)$?
    Answer: $3x^2 - 2x - 4$ This question tests your understanding of how to add polynomials using the key ideas of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (x² - 3x + 2) + (2x² + x - 6) by lining up like terms: the x² terms are x² and 2x², which combine to 3x². The x terms are -3x and x, which combine to -2x. The constant terms are 2 and -6, which combine to -4. Putting it all together: 3x² - 2x - 4. Choice A is correct because it combines all like terms properly, giving us 3x² - 2x - 4 in standard form. Excellent! Choice B combines the coefficients incorrectly, calculating 2 + (-6) as +8 instead of -4. When combining like terms, we add or subtract the coefficients (the numbers), not the exponents! The key to polynomial addition and subtraction: line up your like terms in columns (all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
13. A student claims that $(2x + 3)(x^2 - x + 1) = 2x^3 + x^2 - 2x + 3$. To verify this claim, which of the following approaches would be most efficient?
    Answer: Expand the left side completely and compare all coefficients with the right side To verify polynomial equality, we must compare all corresponding coefficients. Expanding $(2x + 3)(x^2 - x + 1) = 2x^3 - 2x^2 + 2x + 3x^2 - 3x + 3 = 2x^3 + x^2 - x + 3$, which differs from the claimed $2x^3 + x^2 - 2x + 3$ in the coefficient of $x$. Choice A might work by coincidence but doesn't prove equality. Choice B is better but still insufficient since polynomials can agree at finitely many points yet be different. Choice D only checks two coefficients, missing errors in other terms.
14. What is the sum of $(x^3 - 2x^2 + x - 1)$ and $(2x^3 + x^2 - 4x + 5)$?
    Answer: $3x^3 - x^2 - 3x + 4$ This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (x³ - 2x² + x - 1) + (2x³ + x² - 4x + 5) by lining up like terms: the x³ terms are x³ and 2x³, which combine to 3x³. The x² terms are -2x² and x², which combine to -x². The x terms are x and -4x, which combine to -3x. The constant terms are -1 and 5, which combine to 4. Putting it all together: 3x³ - x² - 3x + 4. Choice A is correct because it combines all like terms properly, giving us 3x³ - x² - 3x + 4 in standard form. Excellent! Choice B makes an error when combining the x² terms: -2x² + x² = -x², not -3x². When combining like terms with different signs, we subtract the smaller coefficient from the larger and keep the sign of the larger coefficient! The key to polynomial addition: line up your like terms in columns (all the x³ terms together, all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
15. Subtract: $\left(4x^2 + x - 6\right) - \left(2x^2 - 3x + 5\right)$
    Answer: $2x^2 + 4x - 11$ This question tests your understanding of how to subtract polynomials using the key ideas of distributing the negative sign and combining like terms. Subtracting polynomials works like addition, but with one extra step: first, we distribute the negative sign to every term in the polynomial being subtracted (this flips all the signs), then we combine like terms just like in addition. To subtract (4x² + x - 6) - (2x² - 3x + 5), we first distribute the negative sign: (4x² + x - 6) - (2x² - 3x + 5) = (4x² + x - 6) + (-2x² + 3x - 5). Now we combine like terms: 4x² - 2x² = 2x², x + 3x = 4x, and -6 - 5 = -11, giving us 2x² + 4x - 11. That negative sign distribution is super important—it changes every single sign in the second polynomial! Choice B is correct because it distributes the negative sign to all terms, giving us 2x² + 4x - 11 in standard form. Excellent! Choice A forgets to distribute the negative sign to the middle term. In subtraction, every term in the second polynomial needs to have its sign flipped: -3x becomes +3x, not -3x. It's one of the trickiest parts of polynomial subtraction! When subtracting polynomials, think 'flip then add': flip every single sign in the polynomial you're subtracting (that's distributing the negative), then add like usual. If you see (4x² + x - 6) - (2x² - 3x + 5), it becomes (4x² + x - 6) + (-2x² + 3x - 5) = 2x² + 4x - 11. The sign flip is crucial!
16. Multiply: $(2x - 1)(x + 4)$
    Answer: $2x^2 + 7x - 4$ This question tests your understanding of how to multiply polynomials using the key ideas of distributing and using FOIL. To multiply polynomials, we use the distributive property: each term from the first polynomial gets multiplied by each term from the second polynomial. For binomials, we often use FOIL (First, Outer, Inner, Last) to remember all four products we need to find, then we combine any like terms in the result. Multiplying (2x - 1)(x + 4), we use FOIL: First 2x·x = 2x², Outer 2x·4 = 8x, Inner -1·x = -x, Last -1·4 = -4. Now combine like terms: 2x² + (8x - x) - 4 = 2x² + 7x - 4. Each term from the first binomial gets multiplied by each term from the second! Choice A is correct because it multiplies each term correctly and combines the like terms, giving us 2x² + 7x - 4 in standard form. Excellent! Choice D misses one of the products in the multiplication—an easy thing to do! When multiplying binomials, we need all four products: First, Outer, Inner, and Last. This choice has -7x instead of +7x, perhaps forgetting the outer product or mishandling signs. For multiplication, especially with binomials, FOIL is your friend: First (first terms of each), Outer (outer terms), Inner (inner terms), Last (last terms). Then don't forget to combine any like terms at the end—many students get all four products right but forget this final step!
17. Consider the polynomial multiplication $(x^2 + 2x - 1)(3x^2 - x + 4)$. Without fully expanding, determine which term contributes to the coefficient of $x^3$ in the product.
    Answer: Both $2x \cdot 3x^2$ and $x^2 \cdot (-x)$ contribute to the $x^3$ coefficient To find all terms that contribute to $x^3$ in the product, we identify pairs of terms whose degrees sum to 3. From the first polynomial $(x^2, 2x, -1)$ and second polynomial $(3x^2, -x, 4)$: $2x \cdot 3x^2 = 6x^3$ and $x^2 \cdot (-x) = -x^3$. These are the only two combinations that produce $x^3$ terms. Choice A misses the second contribution. Choice B misses the first contribution. Choice D incorrectly identifies $x^2 \cdot 3x^2 = 3x^4$ (degree 4, not 3) and $(-1) \cdot (-x) = x$ (degree 1, not 3).
18. Find $(5x^2 - 2x + 3) - (x^2 + 4x - 6)$.
    Answer: $4x^2 - 6x + 9$ This question tests your understanding of how to subtract polynomials using the key ideas of distributing the negative and combining like terms. Subtracting polynomials works like addition, but with one extra step: first, we distribute the negative sign to every term in the polynomial being subtracted (this flips all the signs), then we combine like terms just like in addition. To subtract (5x² - 2x + 3) - (x² + 4x - 6), we first distribute the negative sign: (5x² - 2x + 3) + (-x² - 4x + 6). Now we combine like terms: the x² terms are 5x² and -x² to 4x²; the x terms are -2x and -4x to -6x; the constants are 3 and 6 to 9. That negative sign distribution is super important—it changes every single sign in the second polynomial! Choice B is correct because it distributes the negative sign to all terms and combines all like terms properly, giving us 4x² - 6x + 9 in standard form. Excellent! Choice A makes a really common mistake: it forgets to distribute the negative sign to all the terms when subtracting. In subtraction, every term in the second polynomial needs to have its sign flipped: for example, - ( -6) becomes +6. It's one of the trickiest parts of polynomial subtraction! When subtracting polynomials, think 'flip then add': flip every single sign in the polynomial you're subtracting (that's distributing the negative), then add like usual. The sign flip is crucial!
19. Which expression is equivalent to $(x^2 - 4x + 1) + (3x^2 + x - 6)$?
    Answer: $4x^2 - 3x - 5$ This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, $x^2$ and $3x^2$ are like terms (both have $x^2$), so they combine to $4x^2$, but $x^2$ and $x$ are NOT like terms because the exponents are different. Let's add ($x^2$ - $4x$ + 1) + ($3x^2$ + $x$ - 6) by lining up like terms: the $x^2$ terms are $x^2$ and $3x^2$, which combine to $4x^2$. The $x$ terms are $-4x$ and $x$, which combine to $-3x$. The constant terms are 1 and -6, which combine to -5. Putting it all together: $4x^2 - 3x - 5$. Choice A is correct because it combines all like terms properly, giving us $4x^2 - 3x - 5$ in standard form. Excellent! Choice B makes an error with the constant terms: it calculates $1 + (-6) = 7$, but $1 - 6 = -5$. Remember that adding a negative number is the same as subtracting! The key to polynomial addition and subtraction: line up your like terms in columns (all the $x^2$ terms together, all the $x$ terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!
20. A polynomial $T(x)$ has the property that $T(x) + (2x^3 - x + 4) = 3x^3 + x^2 - 2x + 1$. What is the coefficient of $x$ in $T(x)$?
    Answer: $-1$ When you see an equation involving polynomial addition, you need to isolate the unknown polynomial by using subtraction—the inverse operation. Given that $T(x) + (2x^3 - x + 4) = 3x^3 + x^2 - 2x + 1$, you can find $T(x)$ by subtracting the second polynomial from both sides: $T(x) = (3x^3 + x^2 - 2x + 1) - (2x^3 - x + 4)$ Distribute the negative sign carefully: $T(x) = 3x^3 + x^2 - 2x + 1 - 2x^3 + x - 4$ Now combine like terms: - $x^3$ terms: $3x^3 - 2x^3 = x^3$ - $x^2$ terms: $x^2$ - $x$ terms: $-2x + x = -x$ - Constant terms: $1 - 4 = -3$ Therefore, $T(x) = x^3 + x^2 - x - 3$, and the coefficient of $x$ is $-1$. Let's examine why the other answers are wrong: - A) 3: This would come from incorrectly using the constant term $-3$ or mishandling signs - B) 1: This results from forgetting to distribute the negative sign when subtracting $-x$, giving $-2x - x = -3x$ instead of $-2x + x = -x$ - C) -3: This confuses the constant term with the coefficient of $x$ Study tip: When subtracting polynomials, always distribute the negative sign to every term in the second polynomial, then carefully combine like terms. Double-check your work by substituting your answer back into the original equation.

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