The problem is to determine the quotient and remainder when the polynomial $3x^3 - 5x^2 - 7x - 1$ is divided by the polynomial $x - 3$.

1. Problem Description

The problem is to determine the quotient and remainder when the polynomial $3x^3 - 5x^2 - 7x - 1$ is divided by the polynomial $x - 3$.

2. Solution Steps

We will use polynomial long division.

First, we divide $3x^3$ by $x$ to get $3x^2$. Then we multiply $3x^2$ by $(x-3)$ to get $3x^3 - 9x^2$. Subtracting this from $3x^3 - 5x^2$ gives $(3x^3 - 5x^2) - (3x^3 - 9x^2) = 4x^2$. Bring down the $-7x$ term to get $4x^2 - 7x$.

Next, we divide $4x^2$ by $x$ to get $4x$. Then we multiply $4x$ by $(x-3)$ to get $4x^2 - 12x$. Subtracting this from $4x^2 - 7x$ gives $(4x^2 - 7x) - (4x^2 - 12x) = 5x$. Bring down the $-1$ term to get $5x - 1$.

Finally, we divide $5x$ by $x$ to get $5$. Then we multiply $5$ by $(x-3)$ to get $5x - 15$. Subtracting this from $5x - 1$ gives $(5x - 1) - (5x - 15) = 14$.

The quotient is $3x^2 + 4x + 5$ and the remainder is $14$.

$3x^3 - 5x^2 - 7x - 1 = (x-3)(3x^2+4x+5)+14$

3. Final Answer

The quotient is $3x^2 + 4x + 5$ and the remainder is $14$.