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!

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!

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!

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!

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!

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!

This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (2x² - 3x + 8) + (-x² + 6x - 5) by lining up like terms: the x² terms are 2x² and -x², which combine to x². The x terms are -3x and 6x, which combine to 3x. The constant terms are 8 and -5, which combine to 3. Putting it all together: x² + 3x + 3. Choice A is correct because it combines all like terms properly, giving us x² + 3x + 3 in standard form. Excellent! Choice B makes an error when combining the x terms: -3x + 6x = 3x, not 9x. When combining -3x and 6x, think of it as 6x - 3x = 3x, not adding their absolute values! The key to polynomial addition: line up your like terms in columns (all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!

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 (2x² + 3x - 5) + (x² - 4x + 7) by lining up like terms: the x² terms are 2x² and x², which combine to 3x². The x terms are 3x and -4x, which combine to -x. The constant terms are -5 and 7, which combine to 2. Putting it all together: 3x² - x + 2. Choice A is correct because it combines all like terms properly, giving us 3x² - x + 2 in standard form. Excellent! Choice B makes a really common mistake: it combines the coefficients incorrectly, perhaps by miscalculating the x terms as -7x and constants as 12 instead of -x and 2. 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!

This question tests your understanding of how to add polynomials using the key idea of combining like terms. When adding polynomials, we combine like terms—that means adding the coefficients of terms that have the exact same variable parts. For example, 3x² and 5x² are like terms (both have x²), so they combine to 8x², but 3x² and 3x are NOT like terms because the exponents are different. Let's add (3x² - 5x + 7) + (2x² + 4x - 9) by lining up like terms: the x² terms are 3x² and 2x², which combine to 5x². The x terms are -5x and 4x, which combine to -x. The constant terms are 7 and -9, which combine to -2. Putting it all together: 5x² - x - 2. Choice A is correct because it combines all like terms properly, giving us 5x² - x - 2 in standard form. Excellent! Choice B makes a calculation error with the x terms, getting -9x instead of -x. When combining -5x + 4x, we get -1x or just -x, not -9x—remember to add the coefficients carefully! The key to polynomial addition: line up your like terms in columns (all the x² terms together, all the x terms together, all the constants together), then combine each column. This organization prevents mistakes and makes the work so much easier!

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!