Algebra 2 Quiz: Understanding And Operating With Polynomials
20 questions · exam conditions
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Understanding And Operating With PolynomialsQuestion 1 of 20

Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition:

(4x3−2x2+6)+(−x3+5x−9).(4x^3-2x^2+6)+(-x^3+5x-9).

3x3−2x2+5x−33x^3-2x^2+5x-3
3x3+2x2+5x−33x^3+2x^2+5x-3
5x3−2x2+5x−35x^3-2x^2+5x-3
3x3−2x2+5x+153x^3-2x^2+5x+15
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Algebra 2 Quiz

Algebra 2 Quiz: Understanding And Operating With Polynomials

Practice Understanding And Operating With Polynomials in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understanding And Operating With Polynomials, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition:

(4x3−2x2+6)+(−x3+5x−9).(4x^3-2x^2+6)+(-x^3+5x-9).

  1. 3x3−2x2+5x−33x^3-2x^2+5x-3 (correct answer)
  2. 3x3+2x2+5x−33x^3+2x^2+5x-3
  3. 5x3−2x2+5x−35x^3-2x^2+5x-3
  4. 3x3−2x2+5x+153x^3-2x^2+5x+15
Explanation: 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, 4x3−x3=3x34x^3 - x^3 = 3x^3; for x^2, −2x2+0x2=−2x2-2x^2 + 0x^2 = -2x^2; for x, 0x+5x=5x0x + 5x = 5x; for constants, 6−9=−36 - 9 = -3, giving 3x3−2x2+5x−33x^3 - 2x^2 + 5x - 3. Choice A correctly combines all like terms giving 3x3−2x2+5x−33x^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!

Question 2

Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial:

(x2−3x+2)×(x+4).(x^2-3x+2) \times (x+4).

  1. x3+x2−10x+8x^3+x^2-10x+8 (correct answer)
  2. x3+x2−6x+8x^3+x^2-6x+8
  3. x3−7x2−10x+8x^3-7x^2-10x+8
  4. x3+x2−10x−8x^3+x^2-10x-8
Explanation: 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 33 times degree 22 gives degree 55. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute systematically: x2(x+4)=x3+4x2,−3x(x+4)=−3x2−12x,+2(x+4)=2x+8x^2(x + 4) = x^3 + 4x^2,-3x(x + 4) = -3x^2 - 12x,+2(x + 4) = 2x + 8; now combine like terms: x3+(4x2−3x2)+(−12x+2x)+8=x3+x2−10x+8x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8. Choice A correctly distributes completely and combines like terms giving x3+x2−10x+8x^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 33 by degree 22, expect degree 55. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.

Question 3

Simplify and write the result in standard form. The expression uses only addition, subtraction, and multiplication, so by closure the result is a polynomial: (x+2)2−(x−1)(x+3).\left(x+2\right)^2-\left(x-1\right)\left(x+3\right).

  1. x2+2x+7x^2+2x+7
  2. 2x+12x+1
  3. 2x2+72x^2+7
  4. 2x+72x+7 (correct answer)
Explanation: 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. The closure property means that when you add, subtract, or multiply polynomials, you always get a polynomial (never something outside the system). Let's expand each part: (x+2)2=x2+4x+4(x+2)^2 = x^2 + 4x + 4 and (x−1)(x+3)=x2+3x−x−3=x2+2x−3(x-1)(x+3) = x^2 + 3x - x - 3 = x^2 + 2x - 3. Now subtract: (x2+4x+4)−(x2+2x−3)=x2+4x+4−x2−2x+3(x^2 + 4x + 4) - (x^2 + 2x - 3) = x^2 + 4x + 4 - x^2 - 2x + 3. Combining like terms: (x2−x2)+(4x−2x)+(4+3)=0+2x+7=2x+7(x^2 - x^2) + (4x - 2x) + (4 + 3) = 0 + 2x + 7 = 2x + 7. Choice A correctly shows 2x+72x+7 after all operations are performed. Choice B incorrectly includes an x2x^2 term, suggesting the x2x^2 terms weren't properly canceled. The polynomial operation strategy: When simplifying complex expressions, (1) expand all products first, (2) then perform additions/subtractions, (3) combine all like terms, (4) write in standard form. The closure property guarantees your result is always a polynomial!

Question 4

Subtract and write the result in standard form (be careful to distribute the negative sign):  (3x4−x2+6x−9)−(x4+2x3−5x+4) \,(3x^4-x^2+6x-9)-(x^4+2x^3-5x+4)\,.

  1. 2x4−2x3−x2+11x−132x^4-2x^3-x^2+11x-13 (correct answer)
  2. 2x4+2x3−x2+11x−132x^4+2x^3-x^2+11x-13
  3. 2x4−2x3−x2+x−132x^4-2x^3-x^2+x-13
  4. 2x4−2x3+x2+11x−132x^4-2x^3+x^2+11x-13
Explanation: 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: (3x4−x2+6x−9)−(x4+2x3−5x+4)(3x^4-x^2+6x-9)-(x^4+2x^3-5x+4). First distribute the negative: (3x4−x2+6x−9)+(−x4−2x3+5x−4)(3x^4-x^2+6x-9)+(-x^4-2x^3+5x-4). Now combine like terms: x4x^4 terms: 3x4−x4=2x43x^4-x^4=2x^4; x3x^3 terms: 0−2x3=−2x30-2x^3=-2x^3; x2x^2 terms: −x2+0=−x2-x^2+0=-x^2; xx terms: 6x+5x=11x6x+5x=11x; constants: −9−4=−13-9-4=-13. Choice A correctly shows 2x4−2x3−x2+11x−132x^4-2x^3-x^2+11x-13. Choice C has the wrong coefficient for the xx term (+x instead of +11x), likely from computing 6x−(−5x)6x-(-5x) 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).

Question 5

Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition:

(4x3−2x2+6)+(−x3+5x−9).(4x^3-2x^2+6)+( -x^3+5x-9).

  1. 3x3+2x2+5x−33x^3+2x^2+5x-3
  2. 5x3−2x2+5x−35x^3-2x^2+5x-3
  3. 3x3−2x2+5x−33x^3-2x^2+5x-3 (correct answer)
  4. 3x3−2x2+5x+153x^3-2x^2+5x+15
Explanation: 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, 4x3−x3=3x34x^3 - x^3 = 3x^3; for x^2, −2x2+0x2=−2x2-2x^2 + 0x^2 = -2x^2; for x, 0x+5x=5x0x + 5x = 5x; for constants, 6−9=−36 - 9 = -3, giving 3x3−2x2+5x−33x^3 - 2x^2 + 5x - 3. Choice A correctly combines all like terms giving 3x3−2x2+5x−33x^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!

Question 6

Multiply and write the result in standard form (polynomials are closed under multiplication):

(2x3−x+4)(x2−3)(2x^3 - x + 4)(x^2 - 3)

  1. 2x5−7x3+4x2+3x−122x^5 - 7x^3 + 4x^2 + 3x - 12 (correct answer)
  2. 2x5−x3+4x2−6x3+3x−122x^5 - x^3 + 4x^2 - 6x^3 + 3x - 12
  3. 2x5−x3+4x2+3x−122x^5 - x^3 + 4x^2 + 3x - 12
  4. 2x5−7x3+4x2−3x−122x^5 - 7x^3 + 4x^2 - 3x - 12
Explanation: 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 (2x3−x+4)(x2−3)(2x^3 - x + 4)(x^2 - 3), distribute: 2x3(x2−3)=2x5−6x32x^3(x^2 - 3) = 2x^5 - 6x^3, −x(x2−3)=−x3+3x-x(x^2 - 3) = -x^3 + 3x, 4(x2−3)=4x2−124(x^2 - 3) = 4x^2 - 12, then combine: 2x5+(−6x3−x3)+4x2+3x−12=2x5−7x3+4x2+3x−122x^5 + (-6x^3 - x^3) + 4x^2 + 3x - 12 = 2x^5 - 7x^3 + 4x^2 + 3x - 12. Choice B correctly distributes completely and combines like terms giving 2x5−7x3+4x2+3x−122x^5 - 7x^3 + 4x^2 + 3x - 12 in standard form. A common mistake, as in choice A, is not combining all x^3 terms properly, leaving them separate like −x3−6x3-x^3 - 6x^3 instead of −7x3-7x^3. 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.

Question 7

Multiply and write the result in standard form (polynomials are closed under multiplication): (x3−2x2+3)(x−5)(x^3 - 2x^2 + 3)(x - 5)

  1. x4−7x3+10x2+3x−15x^4 - 7x^3 + 10x^2 + 3x - 15 (correct answer)
  2. x4−7x3+10x2+3x+15x^4 - 7x^3 + 10x^2 + 3x + 15
  3. x4−3x3+10x2+3x−15x^4 - 3x^3 + 10x^2 + 3x - 15
  4. x4−7x3+8x2+3x−15x^4 - 7x^3 + 8x^2 + 3x - 15
Explanation: 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 (x3−2x2+3)(x−5)(x^3 - 2x^2 + 3)(x - 5), distribute: x3(x−5)=x4−5x3x^3(x - 5) = x^4 - 5x^3, −2x2(x−5)=−2x3+10x2-2x^2(x - 5) = -2x^3 + 10x^2, 3(x−5)=3x−153(x - 5) = 3x - 15, then combine: x4+(−5x3−2x3)+10x2+3x−15=x4−7x3+10x2+3x−15x^4 + (-5x^3 - 2x^3) + 10x^2 + 3x - 15 = x^4 - 7x^3 + 10x^2 + 3x - 15. Choice A correctly distributes completely and combines like terms giving x4−7x3+10x2+3x−15x^4 - 7x^3 + 10x^2 + 3x - 15 in standard form. A common mistake, as in choice B, is mishandling the constant sign, perhaps adding instead of subtracting to get +15. 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.

Question 8

Simplify and write the result in standard form (this uses polynomial operations and demonstrates closure under ++, −-, and ×\times):

(x+2)2−(x−1)(x+3)(x + 2)^2 - (x - 1)(x + 3)

  1. 2x+72x + 7 (correct answer)
  2. x2+2x+7x^2 + 2x + 7
  3. 2x2+72x^2 + 7
  4. x+7x + 7
Explanation: 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 33 times degree 22 gives degree 55. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! First expand (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4 and (x−1)(x+3)=x2+2x−3(x - 1)(x + 3) = x^2 + 2x - 3, then subtract: (x2+4x+4)−(x2+2x−3)=x2+4x+4−x2−2x+3=2x+7(x^2 + 4x + 4) - (x^2 + 2x - 3) = x^2 + 4x + 4 - x^2 - 2x + 3 = 2x + 7, demonstrating closure through multiple operations. Choice A correctly combines all like terms after expansion and subtraction giving 2x+72x + 7 in standard form. A common error, like in choice B, might involve not subtracting properly and leaving extra terms, such as an unintended x2x^2. 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!

Question 9

Add and write the result in standard form:  (−4x3+2x2+x−8)+(6x3−5x+3) \,(-4x^3+2x^2+x-8)+(6x^3-5x+3)\,. (This demonstrates closure under addition.)

  1. 2x3+2x2−4x−52x^3+2x^2-4x-5 (correct answer)
  2. 2x3+2x2−4x+52x^3+2x^2-4x+5
  3. 2x3+2x2+6x−52x^3+2x^2+6x-5
  4. 10x3+2x2−4x−510x^3+2x^2-4x-5
Explanation: 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: (−4x3+2x2+x−8)+(6x3−5x+3)(-4x^3+2x^2+x-8)+(6x^3-5x+3). Combine like terms: x3x^3 terms: −4x3+6x3=2x3-4x^3+6x^3=2x^3; x2x^2 terms: 2x2+0=2x22x^2+0=2x^2; xx terms: x+(−5x)=−4xx+(-5x)=-4x; constant terms: −8+3=−5-8+3=-5. Choice A correctly combines all like terms giving 2x3+2x2−4x−52x^3+2x^2-4x-5. Choice B has the wrong sign on the constant term (+5 instead of -5), likely from computing −8+3-8+3 as +5+5 instead of −5-5. 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!

Question 10

Multiply and write the result in standard form. This demonstrates the closure property because the product of polynomials is also a polynomial: (x2−3x+2)×(x+4).\left(x^2 - 3x + 2\right) \times \left(x + 4\right).

  1. x3+x2−10x+8x^3+x^2-10x+8 (correct answer)
  2. x3−7x2+14x+8x^3-7x^2+14x+8
  3. x3+x2−6x+8x^3+x^2-6x+8
  4. x3+x2−10x−8x^3+x^2-10x-8
Explanation: 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 (x2−3x+2)(x+4)(x^2-3x+2)(x+4) using distribution: x2(x)+x2(4)−3x(x)−3x(4)+2(x)+2(4)x^2(x)+x^2(4)-3x(x)-3x(4)+2(x)+2(4). This gives: x3+4x2−3x2−12x+2x+8x^3+4x^2-3x^2-12x+2x+8. Combining like terms: x3+(4x2−3x2)+(−12x+2x)+8=x3+x2−10x+8x^3+(4x^2-3x^2)+(-12x+2x)+8 = x^3+x^2-10x+8. Choice A correctly shows x3+x2−10x+8x^3+x^2-10x+8 with all terms properly distributed and combined. Choice B has −7x2-7x^2 and +14x+14x, suggesting errors in distribution or combining. Degree tracking helps verify your work: for multiplication, add the degrees—degree 2 times degree 1 gives degree 3, which matches our result. The closure property guarantees your result is always a polynomial!

Question 11

Add and write the result in standard form (sum of polynomials is a polynomial by closure):

(3x3+2x2−x+5)+(x3−4x2+3x−2)(3x^3 + 2x^2 - x + 5) + (x^3 - 4x^2 + 3x - 2)

  1. 3x3−2x2+2x+33x^3 - 2x^2 + 2x + 3
  2. 4x3−2x2+2x+34x^3 - 2x^2 + 2x + 3 (correct answer)
  3. 4x3−6x2+2x+34x^3 - 6x^2 + 2x + 3
  4. 4x3−2x2+4x+34x^3 - 2x^2 + 4x + 3
Explanation: 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 add these polynomials, group like terms: for x3x^3, 3x3+x3=4x33x^3 + x^3 = 4x^3; for x2x^2, 2x2−4x2=−2x22x^2 - 4x^2 = -2x^2; for xx, −x+3x=2x-x + 3x = 2x; for constants, 5−2=35 - 2 = 3, resulting in 4x3−2x2+2x+34x^3 - 2x^2 + 2x + 3. Choice A correctly combines all like terms giving 4x3−2x2+2x+34x^3 - 2x^2 + 2x + 3 in standard form. A common mistake, as in choice B, is mishandling the x2x^2 terms, perhaps subtracting incorrectly to get −6x2-6x^2 instead of −2x2-2x^2. 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!

Question 12

Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial:

(x2−3x+2)×(x+4).(x^2-3x+2)\times(x+4).

  1. x3+x2−10x+8x^3+x^2-10x+8 (correct answer)
  2. x3−7x2−10x+8x^3-7x^2-10x+8
  3. x3+x2−6x+8x^3+x^2-6x+8
  4. x3+x2−10x−8x^3+x^2-10x-8
Explanation: 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 33 times degree 22 gives degree 55. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute systematically: x2(x+4)=x3+4x2x^2(x + 4) = x^3 + 4x^2, −3x(x+4)=−3x2−12x-3x(x + 4) = -3x^2 - 12x, +2(x+4)=2x+8+2(x + 4) = 2x + 8; now combine like terms: x3+(4x2−3x2)+(−12x+2x)+8=x3+x2−10x+8x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8. Choice A correctly distributes completely and combines like terms giving x3+x2−10x+8x^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 33 by degree 22, expect degree 55. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.

Question 13

Multiply and write the result in standard form. This demonstrates closure under multiplication: (3x3−x+4)×(x2−2). \left(3x^3 - x + 4\right) \times \left(x^2 - 2\right) .

  1. 3x5−6x3+4x2+2x−83x^5-6x^3+4x^2+2x-8
  2. 3x5−7x3−4x2+2x−83x^5-7x^3-4x^2+2x-8
  3. 3x5−7x3+4x2−2x−83x^5-7x^3+4x^2-2x-8
  4. 3x5−7x3+4x2+2x−83x^5-7x^3+4x^2+2x-8 (correct answer)
Explanation: 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 (3x3−x+4)(x2−2)(3x^3 - x + 4)(x^2 - 2) systematically: 3x3(x2)+3x3(−2)−x(x2)−x(−2)+4(x2)+4(−2)3x^3(x^2) + 3x^3(-2) - x(x^2) - x(-2) + 4(x^2) + 4(-2). This gives: 3x5−6x3−x3+2x+4x2−83x^5 - 6x^3 - x^3 + 2x + 4x^2 - 8. Combining like terms: 3x5+(−6x3−x3)+4x2+2x−8=3x5−7x3+4x2+2x−83x^5 + (-6x^3 - x^3) + 4x^2 + 2x - 8 = 3x^5 - 7x^3 + 4x^2 + 2x - 8. Choice A correctly shows 3x5−7x3+4x2+2x−83x^5 - 7x^3 + 4x^2 + 2x - 8 with all terms properly distributed and combined. Choice B has −6x3-6x^3 instead of −7x3-7x^3, missing the combination of −6x3−x3-6x^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!

Question 14

Multiply the polynomials (distribute each term). The result is a polynomial by closure under multiplication:

(x2+2x−1)×(x2−x+3).(x^2+2x-1)\times(x^2-x+3).

  1. x4−x3+0x2+7x−3x^4-x^3+0x^2+7x-3
  2. x4+x3+4x2+7x−3x^4+x^3+4x^2+7x-3
  3. x4+x3+0x2+5x−3x^4+x^3+0x^2+5x-3
  4. x4+x3+0x2+7x−3x^4+x^3+0x^2+7x-3 (correct answer)
Explanation: 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 33 times degree 22 gives degree 55. This multiplicative closure is powerful—you can multiply polynomials indefinitely and always stay within the polynomial system! Distribute term by term: x2(x2−x+3)=x4−x3+3x2x^2(x^2 - x + 3) = x^4 - x^3 + 3x^2, 2x(x2−x+3)=2x3−2x2+6x2x(x^2 - x + 3) = 2x^3 - 2x^2 + 6x, −1(x2−x+3)=−x2+x−3-1(x^2 - x + 3) = -x^2 + x - 3; combine: x4+(−x3+2x3)+(3x2−2x2−x2)+(6x+x)−3=x4+x3+0x2+7x−3x^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 x4+x3+0x2+7x−3x^4 + x^3 + 0x^2 + 7x - 3 in standard form. For example, choice D has +4x2+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 33 by degree 22, expect degree 55. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.

Question 15

Show closure under multiplication by finding the product and writing it in standard form: If P(x)=x2+1P(x)=x^2+1 and Q(x)=x−2Q(x)=x-2, what is P(x)⋅Q(x)P(x)\cdot Q(x)?

  1. x3−2x2+x−2x^3-2x^2+x-2 (correct answer)
  2. x3−2x2−x−2x^3-2x^2-x-2
  3. x3−2x2+1−2x^3-2x^2+1-2
  4. x2−2x+1x^2-2x+1
Explanation: 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)⋅Q(x)=(x2+1)(x−2)P(x) \cdot Q(x) = (x^2+1)(x-2). Distribute each term: x2(x−2)=x3−2x2x^2(x-2) = x^3-2x^2 and 1(x−2)=x−21(x-2) = x-2. Combine all products: x3−2x2+x−2x^3-2x^2+x-2. Choice A correctly shows x3−2x2+x−2x^3-2x^2+x-2. Choice B has the wrong sign on the xx term (-x instead of +x), likely from distributing 1(x−2)1(x-2) as −x−2-x-2 instead of x−2x-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!

Question 16

Multiply and write the result in standard form (the result should be a polynomial by closure):  (2x3−x+3)(x2−2) \,(2x^3-x+3)(x^2-2)\,.

  1. 2x5−5x3+3x2+2x−62x^5-5x^3+3x^2+2x-6 (correct answer)
  2. 2x5−4x3+3x2+2x−62x^5-4x^3+3x^2+2x-6
  3. 2x5−5x3+3x2−2x−62x^5-5x^3+3x^2-2x-6
  4. 2x5−5x3+3x2+2x+62x^5-5x^3+3x^2+2x+6
Explanation: 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 (2x3−x+3)(x2−2)(2x^3-x+3)(x^2-2). Distribute each term: 2x3(x2−2)=2x5−4x32x^3(x^2-2) = 2x^5-4x^3; −x(x2−2)=−x3+2x-x(x^2-2) = -x^3+2x; 3(x2−2)=3x2−63(x^2-2) = 3x^2-6. Combine all products: 2x5−4x3−x3+2x+3x2−6=2x5−5x3+3x2+2x−62x^5-4x^3-x^3+2x+3x^2-6 = 2x^5-5x^3+3x^2+2x-6. Choice A correctly shows 2x5−5x3+3x2+2x−62x^5-5x^3+3x^2+2x-6. Choice C has the wrong sign on the xx term (-2x instead of +2x), likely from computing −x(−2)-x(-2) as −2x-2x instead of +2x+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.

Question 17

Simplify the expression (combine like terms). Because polynomials are closed under addition, subtraction, and multiplication, the simplified result is a polynomial:

(x+2)2−(x−1)(x+3).(x+2)^2-(x-1)(x+3).​

  1. 2x+72x+7 (correct answer)
  2. 2x+12x+1
  3. x2+2x+7x^2+2x+7
  4. −2x+7-2x+7
Explanation: 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). First expand each part: (x + 2)^2 = x^2 + 4x + 4, (x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3; now subtract: x^2 + 4x + 4 - (x2x^2 + 2x - 3) = x^2 + 4x + 4 - x^2 - 2x + 3 = (x2x^2 - x2x^2) + (4x - 2x) + (4 + 3) = 2x + 7. Choice A correctly combines all like terms giving 2x + 7 in standard form. For instance, choice D might result from a sign error in distribution, like forgetting to subtract the entire second expansion—be sure to distribute the negative to every 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). 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!

Question 18

Let P(x)=3x3−2x2+x−5P(x) = 3x^3 - 2x^2 + x - 5 and Q(x)=x3+4x2−3x+1Q(x) = x^3 + 4x^2 - 3x + 1. If R(x)=P(x)−Q(x)+2P(x)R(x) = P(x) - Q(x) + 2P(x), what is the coefficient of x2x^2 in R(x)R(x)?

  1. −10-10 (correct answer)
  2. 22
  3. −6-6
  4. 66
Explanation: First, simplify R(x)=P(x)−Q(x)+2P(x)=3P(x)−Q(x)R(x) = P(x) - Q(x) + 2P(x) = 3P(x) - Q(x). Then 3P(x)=9x3−6x2+3x−153P(x) = 9x^3 - 6x^2 + 3x - 15 and 3P(x)−Q(x)=(9x3−6x2+3x−15)−(x3+4x2−3x+1)=8x3−10x2+6x−163P(x) - Q(x) = (9x^3 - 6x^2 + 3x - 15) - (x^3 + 4x^2 - 3x + 1) = 8x^3 - 10x^2 + 6x - 16. The coefficient of x2x^2 is −10-10. Choice B results from incorrectly computing P(x)−Q(x)P(x) - Q(x) only. Choice C comes from computing 3P(x)3P(x) but forgetting to subtract Q(x)Q(x). Choice D results from a sign error when subtracting Q(x)Q(x).

Question 19

Let S={x2+1,2x−3,x3−x,4}S = \{x^2 + 1, 2x - 3, x^3 - x, 4\}. If we define a new operation ⊕\oplus such that for any two polynomials p,q∈Sp, q \in S, the result p⊕qp \oplus q is also in SS, which operation could ⊕\oplus represent?

  1. Polynomial addition, since the sum of any two polynomials in SS yields another polynomial in SS.
  2. Polynomial multiplication, since the product of any two polynomials in SS yields another polynomial in SS.
  3. Neither addition nor multiplication, since SS is not closed under standard polynomial operations. (correct answer)
  4. Both addition and multiplication, since SS contains polynomials of different degrees and constants.
Explanation: To check closure under addition: (x2+1)+(2x−3)=x2+2x−2(x^2 + 1) + (2x - 3) = x^2 + 2x - 2, which is not in SS. To check closure under multiplication: (x2+1)⋅4=4x2+4(x^2 + 1) \cdot 4 = 4x^2 + 4, which is not in SS. Since SS is not closed under either standard polynomial operation, ⊕\oplus cannot represent addition or multiplication. Choice A incorrectly assumes closure under addition. Choice B incorrectly assumes closure under multiplication. Choice D incorrectly assumes both operations work despite the presence of different degree polynomials.

Question 20

Multiply and write the result in standard form (polynomials are closed under multiplication): (x2−3x+2)(x+4)(x^2 - 3x + 2)(x + 4)

  1. x3+x2−10x+8x^3 + x^2 - 10x + 8 (correct answer)
  2. x3+x2−6x+8x^3 + x^2 - 6x + 8
  3. x3+x2+10x+8x^3 + x^2 + 10x + 8
  4. x3−x2−10x+8x^3 - x^2 - 10x + 8
Explanation: 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 (x2−3x+2)(x+4)(x^2 - 3x + 2)(x + 4), distribute: x2(x+4)=x3+4x2x^2(x + 4) = x^3 + 4x^2, −3x(x+4)=−3x2−12x-3x(x + 4) = -3x^2 - 12x, +2(x+4)=2x+8+2(x + 4) = 2x + 8, then combine like terms: x3+(4x2−3x2)+(−12x+2x)+8=x3+x2−10x+8x^3 + (4x^2 - 3x^2) + (-12x + 2x) + 8 = x^3 + x^2 - 10x + 8. Choice A correctly distributes completely and combines like terms giving x3+x2−10x+8x^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-6x instead of −10x-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 33 by degree 22, expect degree 55. If your result has the wrong degree, you've made an error—go back and check! This degree check catches many mistakes.