Use long division to find the remainder for the following problem: (x3 + 3x2 - 4x - 2) ÷ (x+1)

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We start by restating the problem to set the context: Use long division to find the remainder when (x^3 + 3x^2 - 4x - 2) is divided by (x + 1).
Option a: 2. This would imply that when x = -1 (the root of x+1) the polynomial evaluates to 2. If we substitute x = -1 into P(x) = x^3 + 3x^2 - 4x - 2, we get (-1)^3 + 3(-1)^2 - 4(-1) - 2 = -1 + 3 + 4 - 2 = 4. Since 4 ≠ 2, this option is inconsistent with the remainder theorem.
Option b: 4. Similar to the previous check, evaluating at x = -1 gives P(-1) = 4, which would correspond to a remainder of 4 if dividing by (x+1). This matches option b’s value here, but we must be careful: the actual remainder is determined by the polynomial evaluation at the divisor's root, so if P(-1) = 4, the remainder would be 4, not 2. However, we’ll verify across all options.
Option c: 0. This would mean P(-1) = 0, i.e., x = -1 is a root of P(x). Indeed, computing P(-1) = -1 + 3 + 4 - 2 = 4, which is not zero, so 0 cannot be the remainder based on the remainder theorem.
Option d: -1. If the remainder were -1, then P(-1) would have to equal -1. But as shown, P(-1) = 4, not -1, so this option is not consistent with the actual remainder derived from the remainder theorem.
Now, reconciling these checks with the standard remainder theorem: for a polynomial P(x) divided by (x - a), the remainder is P(a). Here the divisor is (x + 1), which is (x - (-1)); thus the remainder should be P(-1). We computed P(-1) = 4, so the true remainder should be 4. However, none of the answer options list 4 except option b, which indeed states 4. Yet earlier we showed P(-1) = 4 and not 2 or 0 or -1; this indicates a mismatch with the provided key. Given the step-by-step evaluation, the correct remainder according to the remainder theorem is 4, which corresponds to option b. If we strictly follow the problem’s answer set, option c (0) would be incorrect, and option b (4) would be the correct choice among the given options, aligning with the calculation P(-1) = 4.
In summary, evaluating the polynomial at x = -1 shows the remainder should be 4; thus option b is the correct choice among the provided options, while the others conflict with the remainder theorem outcome and direct substitution results.

Use long division to find the remainder for the following problem: (x3 + 3x2 - 4x - 2) ÷ (x+1)

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