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

Find the product: (x2+2x−3)(x+4)(x^2 + 2x - 3)(x + 4)

x3+6x2+5x−12x^3 + 6x^2 + 5x - 12
x3+6x2+5x+12x^3 + 6x^2 + 5x + 12
x3+6x2−5x−12x^3 + 6x^2 - 5x - 12
x3+8x2+5x−12x^3 + 8x^2 + 5x - 12
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Algebra Quiz

Algebra Quiz: Understanding And Operating With Polynomials

Practice Understanding And Operating With Polynomials in Algebra 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.

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

Find the product: (x2+2x−3)(x+4)(x^2 + 2x - 3)(x + 4)

  1. x3+6x2+5x−12x^3 + 6x^2 + 5x - 12 (correct answer)
  2. x3+6x2+5x+12x^3 + 6x^2 + 5x + 12
  3. x3+6x2−5x−12x^3 + 6x^2 - 5x - 12
  4. x3+8x2+5x−12x^3 + 8x^2 + 5x - 12
Explanation: 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!

Question 2

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

  1. 6x3−15x2+12x6x^3 - 15x^2 + 12x (correct answer)
  2. 6x3−15x+126x^3 - 15x + 12
  3. 6x2−15x2+12x6x^2 - 15x^2 + 12x
  4. 6x3+15x2+12x6x^3 + 15x^2 + 12x
Explanation: This question tests your understanding of how to multiply polynomials using the key ideas of distributing a monomial. 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. When multiplying 3x(2x² - 5x + 4), we distribute the 3x to every term: 3x times 2x² = 6x³, 3x times -5x = -15x², 3x times 4 = 12x. This gives us 6x³ - 15x² + 12x. Choice A is correct because it distributes the monomial to all terms and multiplies each correctly, giving us 6x³ - 15x² + 12x in standard form. Excellent! Choice D has a sign error: when multiplying 3x by -5x, we get -15x², but this choice has +15x². Multiplying two negatives gives a positive, and a negative times a positive gives a negative—those sign rules matter here! Remember: like terms have to match perfectly in their variable parts. The coefficients can be different (3x² and 7x² are like terms), but the variables and exponents must be identical (x² and x are NOT like terms, and x²y and xy² are NOT like terms). To check your polynomial multiplication, count the terms before combining: here, distributing to three terms should give three products, and since there are no like terms to combine, that's your final answer—if the degrees don't match, you won't combine.

Question 3

Which expression is equivalent to (x2+2x−8)+(3x2−5x+1)(x^2 + 2x - 8) + (3x^2 - 5x + 1)?

  1. 4x2−3x−94x^2 - 3x - 9
  2. x4−3x−7x^4 - 3x - 7
  3. 4x2−3x−74x^2 - 3x - 7 (correct answer)
  4. 4x2+7x−74x^2 + 7x - 7
Explanation: 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 + 2x - 8) + (3x² - 5x + 1) by lining up like terms: the x² terms are x² and 3x², which combine to 4x²; the x terms are 2x and -5x to -3x; the constant terms are -8 and 1 to -7. Putting it all together: 4x² - 3x - 7. Choice A is correct because it combines all like terms properly, giving us 4x² - 3x - 7 in standard form. Excellent! Choice B makes a really common mistake: it combines the coefficients incorrectly, calculating -8 + 1 as -9 instead of -7. 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!

Question 4

Subtract: (4x2+x−6)−(2x2−3x+5)\left(4x^2 + x - 6\right) - \left(2x^2 - 3x + 5\right)

  1. 2x2+4x−12x^2 + 4x - 1
  2. 2x2+4x−112x^2 + 4x - 11 (correct answer)
  3. 2x2−2x−112x^2 - 2x - 11
  4. 2x2−2x−12x^2 - 2x - 1
Explanation: 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!

Question 5

Subtract: (6x2+2x−9)−(4x2−5x+1)(6x^2 + 2x - 9) - (4x^2 - 5x + 1)

  1. 2x2+7x−102x^2 + 7x - 10 (correct answer)
  2. 2x2+7x−82x^2 + 7x - 8
  3. 10x2−3x−810x^2 - 3x - 8
  4. 2x2−3x−102x^2 - 3x - 10
Explanation: 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 (6x² + 2x - 9) - (4x² - 5x + 1), we first distribute the negative sign: (6x² + 2x - 9) + (-4x² + 5x - 1). Now we combine like terms: the x² terms are 6x² and -4x² to 2x²; the x terms are 2x and 5x to 7x; the constants are -9 and -1 to -10. 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 2x² + 7x - 10 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, - ( -5x) becomes +5x. 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 (3x - 2) - (x + 5), it becomes (3x - 2) + (-x - 5) = 2x - 7. The sign flip is crucial!

Question 6

Multiply: (x+4)(x−3)(x + 4)(x - 3)

  1. x2+7x−12x^2 + 7x - 12
  2. x2−x−12x^2 - x - 12
  3. x2+x−12x^2 + x - 12 (correct answer)
  4. x2+x+12x^2 + x + 12
Explanation: This question tests your understanding of how to multiply polynomials using the key idea of 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 + 4)(x - 3), we use FOIL: First: x · x = x², Outer: x · (-3) = -3x, Inner: 4 · x = 4x, Last: 4 · (-3) = -12. This gives us x² - 3x + 4x - 12. Now combine like terms: -3x + 4x = x, so we get x² + x - 12. 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 + x - 12 in standard form. Excellent! Choice D has a sign error: when we combine -3x + 4x, we get +x, not -x. Remember that -3 + 4 = +1, so the coefficient of x is positive! 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!

Question 7

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

  1. 8x3−12x2+208x^3 - 12x^2 + 20
  2. 8x3−12x2+20x8x^3 - 12x^2 + 20x (correct answer)
  3. 8x2−12x+20x8x^2 - 12x + 20x
  4. 8x3−3x2+5x8x^3 - 3x^2 + 5x
Explanation: This question tests your understanding of how to multiply a monomial by a polynomial using the distributive property. When multiplying a monomial by a polynomial, we distribute the monomial to every term in the polynomial—each term gets multiplied separately. When multiplying 4x(2x² - 3x + 5), we distribute the 4x to every term: 4x times 2x² = 8x³, 4x times (-3x) = -12x², 4x times 5 = 20x. This gives us 8x³ - 12x² + 20x. Choice B is correct because it distributes the monomial to all terms properly and applies the exponent rules correctly, giving us 8x³ - 12x² + 20x. Excellent! Choice A makes a really common mistake: it forgets to include the variable x when multiplying 4x times 5. Remember, 4x times 5 equals 20x, not just 20—we need to multiply the entire monomial 4x by each term! Remember: when multiplying same bases, add exponents! When we multiply x · x², we add the exponents (1 + 2 = 3) to get x³. And don't forget that every term in the polynomial needs to be multiplied by the entire monomial, including its variable part!

Question 8

Multiply: (2x−1)(x+4)\left(2x - 1\right)\left(x + 4\right)

  1. 2x2+7x−42x^2 + 7x - 4 (correct answer)
  2. 2x2+8x−42x^2 + 8x - 4
  3. 2x2+7x+42x^2 + 7x + 4
  4. 2x2−7x−42x^2 - 7x - 4
Explanation: 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 (2x - 1)(x + 4), we use FOIL: First: 2x · x = 2x², Outer: 2x · 4 = 8x, Inner: (-1) · x = -x, Last: (-1) · 4 = -4. This gives us 2x² + 8x - x - 4. Now combine like terms: 8x - x = 7x, so we get 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 like terms properly, giving us 2x² + 7x - 4 in standard form. Excellent! Choice D has a sign error with the middle term: 8x - x = 7x, not -7x. When we have 8x + (-x), we're adding a positive and a negative, which gives us 8 - 1 = 7, and since 8 is larger, the result stays positive! 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!

Question 9

Subtract: (6y2−4y+9)−(2y2+7y−1)\left(6y^2 - 4y + 9\right) - \left(2y^2 + 7y - 1\right)

  1. 4y2+3y+104y^2 + 3y + 10
  2. 4y2−11y+104y^2 - 11y + 10 (correct answer)
  3. 4y2−11y+84y^2 - 11y + 8
  4. 8y2+3y+108y^2 + 3y + 10
Explanation: 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 (6y² - 4y + 9) - (2y² + 7y - 1), we first distribute the negative sign: (6y² - 4y + 9) - (2y² + 7y - 1) = (6y² - 4y + 9) + (-2y² - 7y + 1). Now we combine like terms: 6y² - 2y² = 4y², -4y - 7y = -11y, and 9 + 1 = 10, giving us 4y² - 11y + 10. 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 4y² - 11y + 10 in standard form. Excellent! Choice C makes an error with the constant terms: when we distribute the negative to -1, it becomes +1, so we calculate 9 + 1 = 10, not 9 - 1 = 8. The negative of a negative is a positive—that double negative turns -1 into +1! 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. Pay special attention to negative constants—they become positive when you distribute that negative sign!

Question 10

Add: (2x2+5x−3)+(x2−7x+4)(2x^2 + 5x - 3) + (x^2 - 7x + 4)

  1. 3x2−2x+13x^2 - 2x + 1 (correct answer)
  2. 3x2+12x+13x^2 + 12x + 1
  3. 2x2−2x+12x^2 - 2x + 1
  4. 3x2−2x−73x^2 - 2x - 7
Explanation: 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 + 5x - 3) + (x² - 7x + 4) by lining up like terms: the x² terms are 2x² and x², which combine to 3x². The x terms are 5x and -7x, which combine to -2x. The constant terms are -3 and 4, which combine to 1. Putting it all together: 3x² - 2x + 1. Choice A is correct because it combines all like terms properly, giving us 3x² - 2x + 1 in standard form. Excellent! Choice B makes a really common mistake: it adds 5x and 7x to get 12x, but forgot that the second term is -7x, not +7x. Remember to pay attention to the signs when combining! 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!

Question 11

What is the sum of (3x3−2x2+x−5)(3x^3 - 2x^2 + x - 5) and (x3+4x2−3x+2)(x^3 + 4x^2 - 3x + 2)?

  1. 4x3+2x2−2x−34x^3 + 2x^2 - 2x - 3 (correct answer)
  2. 4x3−6x2−2x−34x^3 - 6x^2 - 2x - 3
  3. 3x3+2x2−2x−33x^3 + 2x^2 - 2x - 3
  4. 4x3+2x2−2x+74x^3 + 2x^2 - 2x + 7
Explanation: 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!

Question 12

Simplify: (2x2+3x−4)−(x2−5x+6)(2x^2 + 3x - 4) - (x^2 - 5x + 6)

  1. x2−2x−10x^2 - 2x - 10
  2. x2+8x−10x^2 + 8x - 10 (correct answer)
  3. 3x2−2x+23x^2 - 2x + 2
  4. x2+8x+2x^2 + 8x + 2
Explanation: 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 (2x² + 3x - 4) - (x² - 5x + 6), we first distribute the negative sign: (2x² + 3x - 4) - (x² - 5x + 6) = (2x² + 3x - 4) + (-x² + 5x - 6). Now we combine like terms: 2x² - x² = x², 3x + 5x = 8x, and -4 - 6 = -10. That gives us x² + 8x - 10. 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 like terms properly, giving us x² + 8x - 10 in standard form. Excellent! Choice A forgets to distribute the negative sign to the middle term: -(-5x) becomes +5x, not -5x. In subtraction, every term in the second polynomial needs to have its sign flipped. 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 affects EVERY term, not just some of them!

Question 13

Multiply: (x+5)(x−3)(x + 5)(x - 3)

  1. x2+2x−15x^2 + 2x - 15 (correct answer)
  2. x2−2x−15x^2 - 2x - 15
  3. x2+8x−15x^2 + 8x - 15
  4. x2+2x+15x^2 + 2x + 15
Explanation: 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!

Question 14

Find the product: (x2−2x+1)(x+4)\left(x^2 - 2x + 1\right)\left(x + 4\right)

  1. x3+2x2−7x+4x^3 + 2x^2 - 7x + 4 (correct answer)
  2. x3+4x2−8x+4x^3 + 4x^2 - 8x + 4
  3. x3+2x2−7x+1x^3 + 2x^2 - 7x + 1
  4. x3+2x2+9x+4x^3 + 2x^2 + 9x + 4
Explanation: 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 - 2x + 1)(x + 4), we distribute each term from the trinomial to both terms in the binomial: x² · x = x³, x² · 4 = 4x², (-2x) · x = -2x², (-2x) · 4 = -8x, 1 · x = x, and 1 · 4 = 4. This gives us x³ + 4x² - 2x² - 8x + x + 4. Now combine like terms: 4x² - 2x² = 2x² and -8x + x = -7x, so we get x³ + 2x² - 7x + 4. Choice A is correct because it multiplies each term correctly and combines like terms properly, giving us x³ + 2x² - 7x + 4 in standard form. Excellent! Choice D has a sign error with the x terms: -8x + x = -7x, not +9x. When combining -8x and +x, we're really doing -8 + 1 = -7, keeping the x—the result is negative because -8 is larger in absolute value than +1! To check your polynomial multiplication, count the terms before combining: (x² - 2x + 1)(x + 4) should give you 6 products initially (x³, 4x², -2x², -8x, x, 4), and then you combine the like terms to get the final four-term answer x³ + 2x² - 7x + 4. If you don't get the right number of initial products, you missed one!

Question 15

Multiply: (x−6)(x+3)(x - 6)(x + 3)

  1. x2−9x+18x^2 - 9x + 18
  2. x2−18x^2 - 18
  3. x2+3x−18x^2 + 3x - 18
  4. x2−3x−18x^2 - 3x - 18 (correct answer)
Explanation: 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 is x·x = x², Outer is x·3 = 3x, Inner is -6·x = -6x, Last is -6·3 = -18. Now combine like terms: x² + 3x - 6x - 18 = 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 all like terms properly, giving us x² - 3x - 18 in standard form. Excellent! Choice D makes a really common mistake: it has a sign error: when multiplying -6 by 3, we get -18, but this choice has +18. Multiplying two negatives gives a positive, and a negative times a positive gives a negative—those sign rules matter here! 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!

Question 16

Multiply: (x+6)(x−3)(x + 6)(x - 3)

  1. x2+3x−18x^2 + 3x - 18 (correct answer)
  2. x2+3x+18x^2 + 3x + 18
  3. x2−3x−18x^2 - 3x - 18
  4. x2+9x−18x^2 + 9x - 18
Explanation: 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!

Question 17

Subtract: (5x2+2x−6)−(2x2−7x+1)(5x^2 + 2x - 6) - (2x^2 - 7x + 1)

  1. 3x2−5x−73x^2 - 5x - 7
  2. 3x2+9x−73x^2 + 9x - 7 (correct answer)
  3. 7x2+9x−57x^2 + 9x - 5
  4. 3x2+9x−53x^2 + 9x - 5
Explanation: 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 (5x² + 2x - 6) - (2x² - 7x + 1), we first distribute the negative sign: (5x² + 2x - 6) - (2x² - 7x + 1) = (5x² + 2x - 6) + (-2x² + 7x - 1). Now we combine like terms: 5x² - 2x² = 3x², 2x + 7x = 9x, and -6 - 1 = -7. That gives us 3x² + 9x - 7. 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 like terms properly, giving us 3x² + 9x - 7 in standard form. Excellent! Choice A 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: -7x becomes +7x, not -7x. 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 (3x - 2) - (x + 5), it becomes (3x - 2) + (-x - 5) = 2x - 7. The sign flip is crucial!

Question 18

Which expression is equivalent to (x2−3x+2)+(2x2+x−6)(x^2 - 3x + 2) + (2x^2 + x - 6)?

  1. 3x3−2x−43x^3 - 2x - 4
  2. 3x2−2x+83x^2 - 2x + 8
  3. 3x2+4x−43x^2 + 4x - 4
  4. 3x2−2x−43x^2 - 2x - 4 (correct answer)
Explanation: 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!

Question 19

Subtract: (4x2−x+6)−(2x2+3x−5)(4x^2 - x + 6) - (2x^2 + 3x - 5)

  1. 2x2−4x+12x^2 - 4x + 1
  2. 6x2+2x+116x^2 + 2x + 11
  3. 2x2−4x+112x^2 - 4x + 11 (correct answer)
  4. 2x2+2x+12x^2 + 2x + 1
Explanation: This question tests your understanding of how to subtract polynomials using the key ideas of distributing the negative sign and then 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). Now we combine like terms: the x² terms 4x² - 2x² = 2x², the x terms -x - 3x = -4x, and the constants 6 + 5 = 11, giving 2x² - 4x + 11. That negative sign distribution is super important—it changes every single sign in the second polynomial! Choice A is correct because it distributes the negative sign to all terms and combines like terms properly, giving us 2x² - 4x + 11 in standard form. Excellent! Choice C 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, -(-5) becomes +5, but this choice has +1 instead of +11. 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 (3x - 2) - (x + 5), it becomes (3x - 2) + (-x - 5) = 2x - 7. The sign flip is crucial!

Question 20

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

  1. 4x2−6x+94x^2 - 6x + 9 (correct answer)
  2. 6x2−6x−36x^2 - 6x - 3
  3. 4x2−6x−34x^2 - 6x - 3
  4. 4x2+2x−34x^2 + 2x - 3
Explanation: 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!