<Question>
Find the quotient when $x^6 - 3$ is divided by $x + 1.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the quotient when \( x^6 - 3 \) is divided by \( x + 1 \). Hmm, let me think about how to approach this. 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 the quicker method. Let me recall how synthetic division works.

<prompt_1.1> Set up synthetic division for the polynomial \( x^6 - 3 \) divided by \( x + 1 \) using the root \( x = -1 \). </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, synthetic division requires that the polynomial be written in standard form, with all degrees of x accounted for, even if their coefficients are zero. So, \( x^6 - 3 \) is already a sixth-degree polynomial, but when dividing by \( x + 1 \), I need to set up the coefficients for each term from \( x^6 \) down to the constant term. Let me write out the coefficients: for \( x^6 \), it's 1; \( x^5 \), 0; \( x^4 \), 0; \( x^3 \), 0; \( x^2 \), 0; \( x \), 0; and the constant term, -3. So the coefficients are [1, 0, 0, 0, 0, 0, -3].

The root we're using for synthetic division is the value that makes the divisor zero. Since the divisor is \( x + 1 \), setting that equal to zero gives \( x = -1 \). So we'll use -1 as the value in synthetic division.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, setting up synthetic division:

<prompt_2.1> Perform synthetic division with the root \( x = -1 \) and the coefficients [1, 0, 0, 0, 0, 0, -3]. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
We write down -1 and the coefficients:

-1 | 1   0    0    0    0    0    -3
    |      -1   1   -1    1   -1    1
    -------------------------------
      1  -1    1   -1    1   -1    -2
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that again. Starting with the first coefficient, which is 1. Multiply by -1, add to the next coefficient: 1*(-1) = -1; 0 + (-1) = -1. Then, take that -1, multiply by -1: 1; add to the next coefficient 0: 1. Then multiply by -1: -1; add to next coefficient 0: -1. Multiply by -1: 1; add to next 0: 1. Multiply by -1: -1; add to next 0: -1. Multiply by -1: 1; add to last coefficient -3: -3 +1 = -2.

So the bottom row after division is [1, -1, 1, -1, 1, -1, -2].
The last number, -2, is the remainder, which makes sense because when you divide by \( x + 1 \), the remainder is \( f(-1) \). Let me verify that. If I plug x = -1 into \( x^6 - 3 \), I get \( (-1)^6 - 3 = 1 - 3 = -2 \), which matches the remainder.

So the quotient is the polynomial formed by the first six coefficients: 1x^5 -1x^4 +1x^3 -1x^2 +1x -1, and the remainder is -2. Therefore, when we divide \( x^6 - 3 \) by \( x + 1 \), the quotient is \( x^5 - x^4 + x^3 - x^2 + x - 1 \) with a remainder of -2.

But the question specifically asks for the quotient. So the answer should be \( x^5 - x^4 + x^3 - x^2 + x - 1 \).

Wait a second, let me confirm using polynomial long division just to be sure. Maybe I made a mistake in the synthetic division steps.

Let's do polynomial long division step by step.

<prompt_3.1> Verify the synthetic division result using polynomial long division. </prompt_3.1>
-----

**Final Answer**
The quotient is \boxed{x^5 - x^4 + x^3 - x^2 + x - 1}.
</Plan_3>

<Execution_3>
<execution_3.1>
Divide \( x^6 - 3 \) by \( x + 1 \).

First, divide \( x^6 \) by \( x \), which gives \( x^5 \). Multiply \( x + 1 \) by \( x^5 \), resulting in \( x^6 + x^5 \). Subtract this from the original polynomial:

\( (x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 3) - (x^6 + x^5) = -x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 3 \).

Next term: divide \( -x^5 \) by \( x \), which is \( -x^4 \). Multiply \( x + 1 \) by \( -x^4 \), giving \( -x^5 - x^4 \). Subtract:

\( (-x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 3) - (-x^5 - x^4) = x^4 + 0x^3 + 0x^2 + 0x - 3 \).

Next term: divide \( x^4 \) by \( x \), which is \( x^3 \). Multiply \( x + 1 \) by \( x^3 \), resulting in \( x^4 + x^3 \). Subtract:

\( (x^4 + 0x^3 + 0x^2 + 0x - 3) - (x^4 + x^3) = -x^3 + 0x^2 + 0x - 3 \).

Next term: divide \( -x^3 \) by \( x \), which is \( -x^2 \). Multiply \( x + 1 \) by \( -x^2 \), giving \( -x^3 - x^2 \). Subtract:

\( (-x^3 + 0x^2 + 0x - 3) - (-x^3 - x^2) = x^2 + 0x - 3 \).

Next term: divide \( x^2 \) by \( x \), which is \( x \). Multiply \( x + 1 \) by \( x \), resulting in \( x^2 + x \). Subtract:

\( (x^2 + 0x - 3) - (x^2 + x) = -x - 3 \).

Next term: divide \( -x \) by \( x \), which is \( -1 \). Multiply \( x + 1 \) by \( -1 \), giving \( -x - 1 \). Subtract:

\( (-x - 3) - (-x - 1) = (-3) + 1 = -2 \).

So the quotient from long division is \( x^5 - x^4 + x^3 - x^2 + x - 1 \) with a remainder of -2, which matches the synthetic division result. Therefore, I can be confident that the quotient is correct.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the quotient when $x^6 - 3$ is divided by $x + 1$ is $\boxed{x^5 - x^4 + x^3 - x^2 + x - 1}$.
</Final_answer>
</think>