Okay, so I need to find the quotient when x^6 minus 3 is divided by x plus 1. Hmm, let me think. I remember that when dividing polynomials, we can use polynomial long division or synthetic division. Since the divisor here is x + 1, which is a linear polynomial, synthetic division might be quicker. Let me try that.

First, I need to set up the synthetic division. The root associated with x + 1 is -1, right? Because if x + 1 = 0, then x = -1. So I'll use -1 as the value in synthetic division. Now, the coefficients of the dividend polynomial x^6 - 3. Wait, the dividend is x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 3. So the coefficients are 1, 0, 0, 0, 0, 0, -3. Let me list them out in order: 1, 0, 0, 0, 0, 0, -3.

Now, setting up synthetic division with -1:

-1 | 1   0   0   0   0   0   -3

I'll bring down the first coefficient, which is 1. Then multiply by -1: 1 * (-1) = -1. Add to the next coefficient: 0 + (-1) = -1. Multiply by -1: -1 * (-1) = 1. Add to the next coefficient: 0 + 1 = 1. Multiply by -1: 1 * (-1) = -1. Add to the next coefficient: 0 + (-1) = -1. Multiply by -1: -1 * (-1) = 1. Add to the next coefficient: 0 + 1 = 1. Multiply by -1: 1 * (-1) = -1. Add to the next coefficient: 0 + (-1) = -1. Multiply by -1: -1 * (-1) = 1. Add to the last coefficient: -3 + 1 = -2.

So the result of the synthetic division is coefficients 1, -1, 1, -1, 1, -1, with a remainder of -2. That means the quotient is x^5 - x^4 + x^3 - x^2 + x - 1, and the remainder is -2. But the question asks for the quotient, so the answer should be x^5 - x^4 + x^3 - x^2 + x - 1.

Wait, let me verify this. If I multiply the divisor (x + 1) by the quotient and add the remainder, it should equal the original polynomial. Let's check:

(x + 1)(x^5 - x^4 + x^3 - x^2 + x - 1) + (-2)

First, multiply x by each term in the quotient:

x * x^5 = x^6

x * (-x^4) = -x^5

x * x^3 = x^4

x * (-x^2) = -x^3

x * x = x^2

x * (-1) = -x

Then multiply 1 by each term in the quotient:

1 * x^5 = x^5

1 * (-x^4) = -x^4

1 * x^3 = x^3

1 * (-x^2) = -x^2

1 * x = x

1 * (-1) = -1

So adding all those terms:

x^6 - x^5 + x^4 - x^3 + x^2 - x + x^5 - x^4 + x^3 - x^2 + x - 1

Now combine like terms:

x^6 + (-x^5 + x^5) + (x^4 - x^4) + (-x^3 + x^3) + (x^2 - x^2) + (-x + x) -1

All the middle terms cancel out, leaving x^6 -1. Then add the remainder of -2: x^6 -1 -2 = x^6 -3. Which matches the original polynomial. So the quotient is correct.

Alternatively, maybe I should try polynomial long division just to confirm. Let's see.

Dividing x^6 -3 by x +1.

First term: x^6 divided by x is x^5. Multiply x +1 by x^5: x^6 + x^5. Subtract from the dividend:

(x^6 -3) - (x^6 + x^5) = -x^5 -3

Next term: -x^5 divided by x is -x^4. Multiply x +1 by -x^4: -x^5 -x^4. Subtract:

(-x^5 -3) - (-x^5 -x^4) = x^4 -3

Next term: x^4 divided by x is x^3. Multiply x +1 by x^3: x^4 + x^3. Subtract:

(x^4 -3) - (x^4 +x^3) = -x^3 -3

Next term: -x^3 divided by x is -x^2. Multiply x +1 by -x^2: -x^3 -x^2. Subtract:

(-x^3 -3) - (-x^3 -x^2) = x^2 -3

Next term: x^2 divided by x is x. Multiply x +1 by x: x^2 +x. Subtract:

(x^2 -3) - (x^2 +x) = -x -3

Next term: -x divided by x is -1. Multiply x +1 by -1: -x -1. Subtract:

(-x -3) - (-x -1) = -2

So the quotient is x^5 -x^4 +x^3 -x^2 +x -1 with a remainder of -2. Same result as synthetic division. So that's correct.

Therefore, the quotient is x^5 -x^4 +x^3 -x^2 +x -1.
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