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

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We start by identifying the division problem: divide the polynomial f(x) = x^3 + 3x^2 - 4x - 2 by (x + 1). The remainder theorem tells us that the remainder of a polynomial divided by (x - a) is f(a). Here, (x + 1) is (x - (-1)), so we evaluate f at x = -1 to find the remainder.

Option a: -1. If you substitute x = -1 into the polynomial, you get f(-1) = (-1)^3 + 3(-1)^2 - 4(-1) - 2 = -1 + 3 + 4 - 2 = 4, not -1. This option is inconsistent with the actual remainder.

Option b: 2. Substituting x = -1 again yields f(-1) = 4, so this value does not match the remainder computed by the remainder theorem. This option is incorrect.

Option c: 4. This matches the result of f(-1) = 4, which accurately gives the remainder when dividing by (x + 1) according to the remainder theorem. If you performed long division, you would obtain the same remainder, 4, at the end.

Option d: 0. Since f(-1) equals 4, not 0, this option is incorrect. A remainder of 0 would imply that (x + 1) is a factor of the polynomial, which is not the case here.

In summary, evaluating the polynomial at x = -1 yields 4, so the remainder of the division is 4. This aligns with option c if you were mapping back to the provided choices, but the essential takeaway is the remainder value 4 as given by the remainder theorem.

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

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

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