This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! The Remainder Theorem states that when you divide a polynomial p(x) by (x - a), the remainder is simply p(a)—just substitute a into the polynomial and evaluate! This works because of how polynomial division works: p(x) = (x - a)·q(x) + r, where r is the remainder. Substituting x = a: p(a) = (a - a)·q(a) + r = 0 + r = r. So the remainder r equals p(a)—brilliant! For (x-2), a=2, $p(2)=2^4$ $+m2^2$ -22 +1=16+4m-4+1=13+4m; set to 9: 13+4m=9, 4m=-4, m=-1. Choice A correctly finds m=-1. A distractor like C (1) might come from setting 16+4m-4+1=9 as 13+4m=9 but solving 4m=-4 incorrectly. Using the Remainder Theorem: (1) Identify the divisor (x - a) and extract a (remember: (x + 3) = (x - (-3)), so a = -3), (2) Substitute a for every x in p(x), (3) Calculate carefully (use parentheses for negative values!), (4) That result is your remainder. If it equals 0, (x - a) is a factor! This method is dramatically faster than polynomial long division.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! A special case is the Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0. This means the remainder is zero, so the division is exact with no remainder. We can test potential factors by just evaluating the polynomial—if p(a) = 0, we've found a factor! This beats trial-and-error factoring when testing specific values. For (x + 2) = (x - (-2)), a = -2; p(-2) = $(-2)^3$ + $3*(-2)^2$ - 4*(-2) - 12 = -8 + 3*4 + 8 - 12 = -8 + 12 + 8 - 12, which is (-8 - 12) + (12 + 8) = -20 + 20 = 0. Choice A correctly determines that yes, it is a factor because p(-2) = 0. Choice B says no because p(-2) = 8, which might come from forgetting to include the -12 or mishandling signs, but actually it's 0—keep track of all terms! Sign safety for negatives: when testing (x + 2) = (x - (-2)), you're evaluating at a = -2. Substitute carefully: if p(x) = x³ - 3x + 5, then p(-2) = (-2)³ - 3(-2) + 5 = -8 + 6 + 5 = 3. Use parentheses around negative values to avoid sign errors! This is where most mistakes happen with the Remainder Theorem.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! The Remainder Theorem states that when you divide a polynomial p(x) by (x - a), the remainder is simply p(a)—just substitute a into the polynomial and evaluate! This works because of how polynomial division works: p(x) = (x - a)·q(x) + r, where r is the remainder. Substituting x = a: p(a) = (a - a)·q(a) + r = 0 + r = r. So the remainder r equals p(a)—brilliant! For (x + 2) = (x - (-2)), a = -2; p(-2) = $3*(-2)^3$ - $(-2)^2$ - 7*(-2) + 2 = 3*(-8) - 4 + 14 + 2 = -24 - 4 + 14 + 2, which is (-24 - 4) + (14 + 2) = -28 + 16 = -12. Choice A correctly evaluates p(-2) as -12. Choice B says 12, possibly from changing all signs incorrectly or forgetting the negative in $(-2)^3$, but use parentheses to track signs properly! Sign safety for negatives: when testing (x + 2) = (x - (-2)), you're evaluating at a = -2. Substitute carefully: if p(x) = x³ - 3x + 5, then p(-2) = (-2)³ - 3(-2) + 5 = -8 + 6 + 5 = 3. Use parentheses around negative values to avoid sign errors! This is where most mistakes happen with the Remainder Theorem.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! The Remainder Theorem states that when you divide a polynomial p(x) by (x - a), the remainder is simply p(a)—just substitute a into the polynomial and evaluate! This works because of how polynomial division works: p(x) = (x - a)·q(x) + r, where r is the remainder. Substituting x = a: p(a) = (a - a)·q(a) + r = 0 + r = r. So the remainder r equals p(a)—brilliant! For (x + 1) = (x - (-1)), a = -1; p(-1) = $2*(-1)^4$ - $(-1)^2$ + 5 = 21 - 1 + 5 = 2 - 1 + 5 = 6. Choice A correctly evaluates p(-1) as 6. Choice B says 4, which could happen if you mistakenly computed $(-1)^4$ as -1 instead of 1, leading to 2(-1) -1 +5 = -2 -1 +5 =2, but remember even powers are positive! Sign safety for negatives: when testing (x + 2) = (x - (-2)), you're evaluating at a = -2. Substitute carefully: if p(x) = x³ - 3x + 5, then p(-2) = (-2)³ - 3(-2) + 5 = -8 + 6 + 5 = 3. Use parentheses around negative values to avoid sign errors! This is where most mistakes happen with the Remainder Theorem.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! The Remainder Theorem states that when you divide a polynomial p(x) by (x - a), the remainder is simply p(a)—just substitute a into the polynomial and evaluate! This works because of how polynomial division works: p(x) = (x - a)·q(x) + r, where r is the remainder. Substituting x = a: p(a) = (a - a)·q(a) + r = 0 + r = r. So the remainder r equals p(a)—brilliant! To verify, evaluate p(1): $1^4$ = 1, $-31^2$ = -3, +21 = +2, +1; combine: 1 - 3 = -2, -2 + 2 = 0, 0 + 1 = 1. Choice A correctly verifies it's true because p(1)=1 matches the claimed remainder. Choice B incorrectly claims false with p(1)=0, perhaps from miscalculating 1 -3 +2 +1 as 0 by ignoring the last +1. Using the Remainder Theorem: (1) Identify the divisor (x - a) and extract a (remember: (x + 3) = (x - (-3)), so a = -3), (2) Substitute a for every x in p(x), (3) Calculate carefully (use parentheses for negative values!), (4) That result is your remainder. If it equals 0, (x - a) is a factor! This method is dramatically faster than polynomial long division.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! The Remainder Theorem states that when you divide a polynomial p(x) by (x - a), the remainder is simply p(a)—just substitute a into the polynomial and evaluate! This works because of how polynomial division works: $p(x) = (x - a) \cdot q(x) + r$, where r is the remainder. Substituting x = a: $p(a) = (a - a) \cdot q(a) + r = 0 + r = r$. So the remainder r equals p(a)—brilliant! Here, p(1) = 3 and we set it equal to 3: $1^3 - 2(1)^2 + k \cdot 1 - 5 = 1 - 2 + k - 5 = -6 + k = 3$, so $k = 9$. Choice B correctly finds k=9. A distractor like k=7 in choice A might come from misadding -6 + k =3 as k=-3+6=3 or similar arithmetic error. Using the Remainder Theorem: (1) Identify the divisor (x - a) and extract a (remember: $(x + 3) = (x - (-3))$, so a = -3), (2) Substitute a for every x in p(x), (3) Calculate carefully (use parentheses for negative values!), (4) That result is your remainder. If it equals 0, (x - a) is a factor! This method is dramatically faster than polynomial long division.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! A special case is the Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0. This means the remainder is zero, so the division is exact with no remainder. We can test potential factors by just evaluating the polynomial—if p(a) = 0, we've found a factor! This beats trial-and-error factoring when testing specific values. For (x + 2) = (x - (-2)), a = -2; p(-2) = $(-2)^3$ + $3*(-2)^2$ - 4*(-2) - 12 = -8 + 3*4 + 8 - 12 = -8 + 12 + 8 - 12, which is (-8 - 12) + (12 + 8) = -20 + 20 = 0. Choice A correctly determines that yes, it is a factor because p(-2) = 0. Choice B says no because p(-2) = 8, which might come from forgetting to include the -12 or mishandling signs, but actually it's 0—keep track of all terms! Sign safety for negatives: when testing (x + 2) = (x - (-2)), you're evaluating at a = -2. Substitute carefully: if p(x) = x³ - 3x + 5, then p(-2) = (-2)³ - 3(-2) + 5 = -8 + 6 + 5 = 3. Use parentheses around negative values to avoid sign errors! This is where most mistakes happen with the Remainder Theorem.

This question tests your understanding of the Remainder Theorem—a powerful shortcut that lets you find the remainder when dividing a polynomial by (x - a) without doing any long division! A special case is the Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0. This means the remainder is zero, so the division is exact with no remainder. We can test potential factors by just evaluating the polynomial—if p(a) = 0, we've found a factor! This beats trial-and-error factoring when testing specific values. To check if x = -1 is a zero of p(x) = x³ + 2x² - x - 2, we evaluate p(-1): p(-1) = (-1)³ + 2(-1)² - (-1) - 2 = -1 + 2(1) + 1 - 2 = -1 + 2 + 1 - 2 = 0. Since p(-1) = 0, x = -1 is indeed a zero! Choice A correctly states that x = -1 is a zero because p(-1) = 0. Using the Remainder Theorem: (1) A zero of p(x) is a value a where p(a) = 0, (2) This also means (x - a) is a factor of p(x), (3) Calculate carefully with parentheses for negative values, (4) If p(a) = 0, you've found both a zero and a factor! This connection between zeros and factors is fundamental in algebra.

Since $$p(2) = -5$$, we have $$q(2) = p(2) + 5 = -5 + 5 = 0$$. By the Remainder Theorem, since $$q(2) = 0$$, the factor $$(x - 2)$$ divides $$q(x)$$ evenly. Choice B is incorrect because it uses $$(x + 2)$$ instead of $$(x - 2)$$. Choice C is wrong because $$q(x)$$ has remainder $$0$$, not $$-5$$, when divided by $$(x - 2)$$. Choice D is incorrect because it involves division by $$(x + 2)$$ and the wrong remainder value.

Since $$x^2 + 2x - 3 = (x + 3)(x - 1)$$, when $$f(x)$$ is divided by this quadratic, the remainder must be linear: $$R(x) = ax + b$$. By the Remainder Theorem, $$R(-3) = f(-3) = 8$$ and $$R(1) = f(1) = 0$$. So $$-3a + b = 8$$ and $$a + b = 0$$. Solving: $$b = -a$$ and $$-3a + (-a) = 8$$, so $$-4a = 8$$ and $$a = -2$$, $$b = 2$$. Therefore $$R(x) = -2x + 2$$. The other choices don't satisfy both conditions simultaneously.