Understanding and Operating with Polynomials - Algebra 2

1. Because polynomials are closed under subtraction, the difference of two polynomials is also a polynomial. Subtract and write the result in standard form: $\,(x^3+5x^2-2x+7)-(2x^3-x^2+4x-1)\,$.
   Answer: $-x^3+6x^2-6x+8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. For subtraction, remember to distribute the negative sign to every term of the polynomial being subtracted, then combine like terms just like in addition. Let's subtract: $(x^3+5x^2-2x+7)-(2x^3-x^2+4x-1)$. First distribute the negative: $(x^3+5x^2-2x+7)+(-2x^3+x^2-4x+1)$. Now combine like terms: $x^3$ terms: $x^3-2x^3=-x^3$; $x^2$ terms: $5x^2+x^2=6x^2$; $x$ terms: $-2x-4x=-6x$; constants: $7+1=8$. Choice C correctly shows $-x^3+6x^2-6x+8$. Choice B has the wrong sign on the $x$ term (+2x instead of -6x), likely from forgetting to distribute the negative to the $4x$ term. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). Degree tracking helps verify your work: for addition/subtraction, the result's degree should be the maximum of the input degrees (or less if leading terms cancel).
2. Multiply and write the result in standard form (polynomials are closed under multiplication): $(x^2 - 3x + 2)(x + 4)$
   Answer: $x^3 + x^2 - 10x + 8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. The degree of the product is the sum of the individual degrees: degree 3 times degree 2 gives degree 5. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! To multiply $(x^2 - 3x + 2)(x + 4)$, distribute: $x^2(x + 4) = x^3 + 4x^2$, $-3x(x + 4) = -3x^2 - 12x$, $+2(x + 4) = 2x + 8$, then combine like terms: $x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8$. Choice A correctly distributes completely and combines like terms giving $x^3 + x^2 - 10x + 8$ in standard form. A common mistake, as in choice B, is missing some like-term combinations, such as not fully adding the x terms, leading to $-6x$ instead of $-10x$. Degree tracking helps verify your work: for addition/subtraction, the result's degree should be the maximum of the input degrees (or less if leading terms cancel). For multiplication, add the degrees: if multiplying degree $3$ by degree $2$, expect degree $5$. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.
3. A student claims that for any polynomials $P(x)$ and $Q(x)$, the expression $(P(x) + Q(x))^2$ is always equal to $P(x)^2 + Q(x)^2$. Which pair of simple polynomials provides the smallest counterexample to disprove this claim?
   Answer: $P(x) = x$ and $Q(x) = 1$, since $(x + 1)^2 = x^2 + 2x + 1 \neq x^2 + 1$ The student's claim ignores the cross term $2P(x)Q(x)$ in the expansion of $(P(x) + Q(x))^2 = P(x)^2 + 2P(x)Q(x) + Q(x)^2$. The simplest counterexample uses the lowest-degree polynomials. Option A: $P(x) = x, Q(x) = 1$ gives $(x+1)^2 = x^2 + 2x + 1$ while $P(x)^2 + Q(x)^2 = x^2 + 1$. The difference is $2x$, providing a clear counterexample with minimal complexity. Option B uses higher-degree terms unnecessarily. Option C uses coefficients greater than 1, making it more complex. Option D incorrectly treats polynomials as numbers in the final comparison.
4. What is the degree of the polynomial product? (This uses the fact that polynomials are closed under multiplication and that degrees add when multiplying.) If $P(x)$ has degree $3$ and $Q(x)$ has degree $4$, what is the degree of $P(x)\cdot Q(x)$?
   Answer: $7$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, the degree of the product is the sum of the individual degrees: degree 3 times degree 2 gives degree 5. For this problem, we have $P(x)$ with degree 3 and $Q(x)$ with degree 4. When we multiply them, the highest degree term will come from multiplying the highest degree term of $P(x)$ (which has degree 3) by the highest degree term of $Q(x)$ (which has degree 4). Choice A correctly identifies that $3+4=7$ is the degree of the product. Choice B suggests 12, perhaps from multiplying degrees instead of adding them—a common misconception. Degree tracking helps verify your work: for multiplication, add the degrees. This degree check catches many mistakes and confirms the closure property—the product is always a polynomial with predictable degree!
5. Polynomials are closed under addition, meaning the sum of two polynomials is also a polynomial. Add and write the result in standard form: $\,(2x^4-3x^3+x^2-5)+(x^4+2x^3-4x^2+x+3)\,$.
   Answer: $3x^4-x^3-3x^2+x-2$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. Let's add systematically: $(2x^4-3x^3+x^2-5)+(x^4+2x^3-4x^2+x+3)$. Combine like terms: $x^4$ terms: $2x^4+x^4=3x^4$; $x^3$ terms: $-3x^3+2x^3=-x^3$; $x^2$ terms: $x^2+(-4x^2)=-3x^2$; $x$ terms: $0+x=x$; constant terms: $-5+3=-2$. Choice A correctly combines all like terms giving $3x^4-x^3-3x^2+x-2$. Choice B has the wrong constant term (+8 instead of -2), likely from adding -5+3 incorrectly. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). The closure property guarantees your result is always a polynomial!
6. Polynomials are closed under multiplication, so the product of two polynomials is also a polynomial. Multiply and write the result in standard form: $\,(x^2-3x+2)(x+4)\,$.
   Answer: $x^3+x^2-10x+8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. Let's multiply $(x^2-3x+2)(x+4)$ by distributing each term of the first polynomial: $x^2(x+4) = x^3+4x^2$; $-3x(x+4) = -3x^2-12x$; $2(x+4) = 2x+8$. Now combine all products: $x^3+4x^2-3x^2-12x+2x+8 = x^3+x^2-10x+8$. Choice A correctly shows $x^3+x^2-10x+8$. Choice C has the wrong coefficient for $x^2$ (-7 instead of +1), likely from computing $4x^2-3x^2$ incorrectly. The polynomial operation strategy: For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. The degree of the product is the sum of the individual degrees: degree 2 times degree 1 gives degree 3. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system!
7. Add and write the result in standard form: $\,(-4x^3+2x^2+x-8)+(6x^3-5x+3)\,$. (This demonstrates closure under addition.)
   Answer: $2x^3+2x^2-4x-5$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. Let's add: $(-4x^3+2x^2+x-8)+(6x^3-5x+3)$. Combine like terms: $x^3$ terms: $-4x^3+6x^3=2x^3$; $x^2$ terms: $2x^2+0=2x^2$; $x$ terms: $x+(-5x)=-4x$; constant terms: $-8+3=-5$. Choice A correctly combines all like terms giving $2x^3+2x^2-4x-5$. Choice B has the wrong sign on the constant term (+5 instead of -5), likely from computing $-8+3$ as $+5$ instead of $-5$. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). The closure property guarantees your result is always a polynomial!
8. Add the polynomials (combine like terms). This illustrates the closure property because the sum of two polynomials is also a polynomial: $(2x^4-3x^3+x^2-5)+(x^4+2x^3-4x^2+x+3).$
   Answer: $3x^4-x^3-3x^2+x-2$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. For subtraction, remember to distribute the negative sign to every term of the polynomial being subtracted, then combine like terms just like in addition. The result is always a polynomial with degree at most the maximum of the original degrees (it could be less if highest-degree terms cancel). To solve, group the terms by powers: for $x^4$, $2x^4 + x^4 = 3x^4$; for $x^3$, $-3x^3 + 2x^3 = -x^3$; for $x^2$, $x^2 - 4x^2 = -3x^2$; for $x$, $0x + x = x$; for constants, $-5 + 3 = -2$, resulting in $3x^4 - x^3 - 3x^2 + x - 2$. Choice A correctly combines all like terms giving $3x^4 - x^3 - 3x^2 + x - 2$ in standard form. For example, choice B has a constant of +8, which might occur if you mistakenly added extra terms or flipped signs—always double-check each coefficient. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. The closure property guarantees your result is always a polynomial!
9. Let $P(x) = 3x^3 - 2x^2 + x - 5$ and $Q(x) = x^3 + 4x^2 - 3x + 1$. If $R(x) = P(x) - Q(x) + 2P(x)$, what is the coefficient of $x^2$ in $R(x)$?
   Answer: $-10$ First, simplify $R(x) = P(x) - Q(x) + 2P(x) = 3P(x) - Q(x)$. Then $3P(x) = 9x^3 - 6x^2 + 3x - 15$ and $3P(x) - Q(x) = (9x^3 - 6x^2 + 3x - 15) - (x^3 + 4x^2 - 3x + 1) = 8x^3 - 10x^2 + 6x - 16$. The coefficient of $x^2$ is $-10$. Choice B results from incorrectly computing $P(x) - Q(x)$ only. Choice C comes from computing $3P(x)$ but forgetting to subtract $Q(x)$. Choice D results from a sign error when subtracting $Q(x)$.
10. Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial: $(x^2-3x+2) \times (x+4).$
    Answer: $x^3+x^2-10x+8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. The degree of the product is the sum of the individual degrees: degree $3$ times degree $2$ gives degree $5$. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute systematically: $x^2(x + 4) = x^3 + 4x^2,-3x(x + 4) = -3x^2 - 12x,+2(x + 4) = 2x + 8$; now combine like terms: $x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8$. Choice A correctly distributes completely and combines like terms giving $x^3 + x^2 - 10x + 8$ in standard form. For example, choice B might come from forgetting to combine the $x^2$ terms properly, resulting in an incorrect coefficient—always check for like terms after distributing. Degree tracking helps verify your work: for addition/subtraction, the result's degree should be the maximum of the input degrees (or less if leading terms cancel). For multiplication, add the degrees: if multiplying degree $3$ by degree $2$, expect degree $5$. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.
11. Subtract and write the result in standard form (polynomials are closed under subtraction): $(4x^4 - 2x^2 + x - 9) - (x^4 + 3x^3 - 2x + 5)$
    Answer: $3x^4 - 3x^3 - 2x^2 + 3x - 14$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. For subtraction, remember to distribute the negative sign to every term of the polynomial being subtracted, then combine like terms just like in addition. The result is always a polynomial with degree at most the maximum of the original degrees (it could be less if highest-degree terms cancel). To subtract, distribute the negative: $-(x^4 + 3x^3 - 2x + 5) = -x^4 - 3x^3 + 2x - 5$, then add to the first: $4x^4 - x^4 - 2x^2 + x + 2x - 3x^3 - 9 - 5 = 3x^4 - 3x^3 - 2x^2 + 3x - 14$. Choice A correctly combines all like terms giving $3x^4 - 3x^3 - 2x^2 + 3x - 14$ in standard form. A common mistake, as in choice B, is failing to distribute the negative to the $x^3$ term properly, leading to $+3x^3$ instead of $-3x^3$. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. The closure property guarantees your result is always a polynomial!
12. Consider the polynomials $f(x) = 2x^3 - 3x + 4$ and $g(x) = x^2 - 2x + 1$. If $h(x) = f(x) - 2g(x) + g(x) \cdot f(x)$, what is the degree of $h(x)$?
    Answer: 5 First, $f(x) - 2g(x) = (2x^3 - 3x + 4) - 2(x^2 - 2x + 1) = 2x^3 - 2x^2 + x + 2$, which has degree 3. Next, $g(x) \cdot f(x)$ is the product of a degree 2 polynomial and a degree 3 polynomial, giving degree $2 + 3 = 5$. When we add these: $h(x) = (2x^3 - 2x^2 + x + 2) + g(x) \cdot f(x)$, the highest degree term comes from $g(x) \cdot f(x)$, which is degree 5. Therefore, $h(x)$ has degree 5. Choice A only considers $f(x) - 2g(x)$. Choice C incorrectly multiplies degrees instead of adding them. Choice D results from miscalculating the degree of the product.
13. Subtract and write the result in standard form. Be careful to distribute the negative sign to every term; by closure, the difference is a polynomial: $\left(4x^4 - 2x^2 + 3x - 9\right) - \left(x^4 + 5x^3 - 2x + 6\right) .$
    Answer: $3x^4-5x^3-2x^2+5x-15$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When subtracting polynomials, remember to distribute the negative sign to every term of the polynomial being subtracted, then combine like terms just like in addition. Let's subtract: $ ($4x^4$ - $2x^2$ + 3x - 9) - ($x^4$ + $5x^3$ - 2x + 6) $. Distribute the negative: $ ($4x^4$ - $2x^2$ + 3x - 9) + ($-x^4$ - $5x^3$ + 2x - 6) $. Now combine like terms: $ $4x^4$ - $x^4$ = $3x^4$ $; $ 0 - $5x^3$ = -$5x^3$ $; $ -$2x^2$ + 0 = -$2x^2$ $; $ 3x + 2x = 5x $; $ -9 - 6 = -15 $. Choice A correctly shows $3x^4-5x^3-2x^2+5x-15$ with all signs properly handled. Choice B has $+5x^3$ instead of $-5x^3$, a classic sign error when distributing the negative. The polynomial operation strategy: For subtraction: (1) distribute the negative to every term in the second polynomial first, (2) identify all like terms, (3) combine coefficients carefully, (4) write in standard form. This systematic approach prevents sign errors!
14. Show closure under multiplication by finding the product and writing it in standard form: If $P(x)=x^2+1$ and $Q(x)=x-2$, what is $P(x)\cdot Q(x)$?
    Answer: $x^3-2x^2+x-2$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. Let's find $P(x) \cdot Q(x) = (x^2+1)(x-2)$. Distribute each term: $x^2(x-2) = x^3-2x^2$ and $1(x-2) = x-2$. Combine all products: $x^3-2x^2+x-2$. Choice A correctly shows $x^3-2x^2+x-2$. Choice B has the wrong sign on the $x$ term (-x instead of +x), likely from distributing $1(x-2)$ as $-x-2$ instead of $x-2$. The polynomial operation strategy: For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. The degree of the product is the sum of the individual degrees: degree 2 times degree 1 gives degree 3. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system!
15. Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial: $(x^2-3x+2)\times(x+4).$
    Answer: $x^3+x^2-10x+8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. The degree of the product is the sum of the individual degrees: degree $3$ times degree $2$ gives degree $5$. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute systematically: $x^2(x + 4) = x^3 + 4x^2$, $-3x(x + 4) = -3x^2 - 12x$, $+2(x + 4) = 2x + 8$; now combine like terms: $x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8$. Choice A correctly distributes completely and combines like terms giving $x^3 + x^2 - 10x + 8$ in standard form. For example, choice B might come from forgetting to combine the $x^2$ terms properly, resulting in an incorrect coefficient—always check for like terms after distributing. Degree tracking helps verify your work: for addition/subtraction, the result's degree should be the maximum of the input degrees (or less if leading terms cancel). For multiplication, add the degrees: if multiplying degree $3$ by degree $2$, expect degree $5$. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.
16. A polynomial $P(x)$ has degree 4, and when $P(x)$ is added to the polynomial $3x^3 - 2x^2 + x - 5$, the result is a polynomial of degree 3. Which statement about $P(x)$ must be true?
    Answer: The coefficient of $x^4$ in $P(x)$ is nonzero and the coefficient of $x^3$ in $P(x)$ is $-3$. For $P(x)$ to have degree 4, its coefficient of $x^4$ must be nonzero. When adding $P(x)$ to $3x^3 - 2x^2 + x - 5$, the sum has degree 3, meaning the $x^4$ term disappears but the $x^3$ term remains. This is only possible if the $x^3$ coefficient in $P(x)$ is $-3$ (to cancel with the $+3x^3$ from the given polynomial, leaving a nonzero $x^3$ term in the sum). Choice A discusses irrelevant coefficients. Choice B incorrectly states the $x^4$ coefficient is zero. Choice C doesn't specify the required relationship between coefficients.
17. Multiply the polynomials (distribute each term). The result is a polynomial by closure under multiplication: $(x^2+2x-1)\times(x^2-x+3).$
    Answer: $x^4+x^3+0x^2+7x-3$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. The degree of the product is the sum of the individual degrees: degree $3$ times degree $2$ gives degree $5$. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute term by term: $x^2(x^2 - x + 3) = x^4 - x^3 + 3x^2$, $2x(x^2 - x + 3) = 2x^3 - 2x^2 + 6x$, $-1(x^2 - x + 3) = -x^2 + x - 3$; combine: $x^4 + (-x^3 + 2x^3) + (3x^2 - 2x^2 - x^2) + (6x + x) - 3 = x^4 + x^3 + 0x^2 + 7x - 3$. Choice A correctly distributes completely giving $x^4 + x^3 + 0x^2 + 7x - 3$ in standard form. For example, choice D has $+4x^2$, possibly from not combining the $x^2$ terms properly—always add coefficients of like terms carefully. Degree tracking helps verify your work: for addition/subtraction, the result's degree should be the maximum of the input degrees (or less if leading terms cancel). For multiplication, add the degrees: if multiplying degree $3$ by degree $2$, expect degree $5$. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.
18. Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition: $(4x^3-2x^2+6)+( -x^3+5x-9).$
    Answer: $3x^3-2x^2+5x-3$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When adding or subtracting polynomials, the key is combining like terms—terms with exactly the same variable parts. For subtraction, remember to distribute the negative sign to every term of the polynomial being subtracted, then combine like terms just like in addition. The result is always a polynomial with degree at most the maximum of the original degrees (it could be less if highest-degree terms cancel). Combine like terms directly since it's addition: for $x^3$, $4x^3 - x^3 = 3x^3$; for $x^2$, $-2x^2 + 0x^2 = -2x^2$; for x, $0x + 5x = 5x$; for constants, $6 - 9 = -3$, giving $3x^3 - 2x^2 + 5x - 3$. Choice A correctly combines all like terms giving $3x^3 - 2x^2 + 5x - 3$ in standard form. For instance, choice B has +15 for the constant, which could happen if you added instead of subtracting the constants—always verify the operation. The polynomial operation strategy: For addition/subtraction: (1) if subtracting, distribute the negative to every term in the second polynomial first, (2) identify all like terms (same variable parts), (3) combine coefficients of like terms, (4) write in standard form (descending powers). For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. The closure property guarantees your result is always a polynomial!
19. Multiply and write the result in standard form (the result should be a polynomial by closure): $\,(2x^3-x+3)(x^2-2)\,$.
    Answer: $2x^5-5x^3+3x^2+2x-6$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. Let's multiply $(2x^3-x+3)(x^2-2)$. Distribute each term: $2x^3(x^2-2) = 2x^5-4x^3$; $-x(x^2-2) = -x^3+2x$; $3(x^2-2) = 3x^2-6$. Combine all products: $2x^5-4x^3-x^3+2x+3x^2-6 = 2x^5-5x^3+3x^2+2x-6$. Choice A correctly shows $2x^5-5x^3+3x^2+2x-6$. Choice C has the wrong sign on the $x$ term (-2x instead of +2x), likely from computing $-x(-2)$ as $-2x$ instead of $+2x$. The polynomial operation strategy: For multiplication: (1) distribute each term of first polynomial to each term of second, (2) combine any like terms in the result, (3) write in standard form. Degree tracking helps verify your work: for multiplication, add the degrees—if multiplying degree 3 by degree 2, expect degree 5.
20. Multiply and write the result in standard form. This demonstrates closure under multiplication: $\left(3x^3 - x + 4\right) \times \left(x^2 - 2\right) .$
    Answer: $3x^5-7x^3+4x^2+2x-8$ This question tests your understanding that polynomials form a system analogous to integers—they're closed under addition, subtraction, and multiplication, meaning these operations on polynomials always produce another polynomial. When multiplying polynomials, every term from the first polynomial must be distributed to every term of the second polynomial, creating many products that you then combine. Let's multiply $ ($3x^3$ - x + 4)($x^2$ - 2) $ systematically: $3x^3(x^2) + 3x^3(-2) - x(x^2) - x(-2) + 4(x^2) + 4(-2)$. This gives: $3x^5 - 6x^3 - x^3 + 2x + 4x^2 - 8$. Combining like terms: $3x^5 + (-6x^3 - x^3) + 4x^2 + 2x - 8 = 3x^5 - 7x^3 + 4x^2 + 2x - 8$. Choice A correctly shows $3x^5 - 7x^3 + 4x^2 + 2x - 8$ with all terms properly distributed and combined. Choice B has $-6x^3$ instead of $-7x^3$, missing the combination of $-6x^3 - x^3$. Degree tracking helps verify: degree 3 times degree 2 gives degree 5, which matches our highest term. The closure property guarantees your result is always a polynomial!

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