Divide p(x) by g(x) in each of the following questions and find the quotient q(x) and remainder r(x) :
`p(x)=x^(3)-7x^(2)-6x+1, " " g(x)=x-3`

Answer

To divide the polynomial \( p(x) = x^3 - 7x^2 - 6x + 1 \) by \( g(x) = x - 3 \), we will use polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: Write \( p(x) \) as the dividend and \( g(x) \) as the divisor.
\[ \text{Dividend: } x^3 - 7x^2 - 6x + 1 \]
\[ \text{Divisor: } x - 3 \]

2. **Divide the leading terms**: Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \).
\[ \frac{x^3}{x} = x^2 \]
This gives us the first term of the quotient \( q(x) \).

3. **Multiply and subtract**: Multiply \( x^2 \) by the entire divisor \( x - 3 \) and subtract from the dividend.
\[ x^2(x - 3) = x^3 - 3x^2 \]
Now subtract:
\[ (x^3 - 7x^2) - (x^3 - 3x^2) = -7x^2 + 3x^2 = -4x^2 \]
Bring down the next term from the dividend:
\[ -4x^2 - 6x + 1 \]

4. **Repeat the process**: Now divide the leading term \( -4x^2 \) by \( x \).
\[ \frac{-4x^2}{x} = -4x \]
Multiply and subtract:
\[ -4x(x - 3) = -4x^2 + 12x \]
Subtract:
\[ (-4x^2 - 6x) - (-4x^2 + 12x) = -6x - 12x = -18x \]
Bring down the next term:
\[ -18x + 1 \]

5. **Continue the process**: Divide \( -18x \) by \( x \).
\[ \frac{-18x}{x} = -18 \]
Multiply and subtract:
\[ -18(x - 3) = -18x + 54 \]
Subtract:
\[ (-18x + 1) - (-18x + 54) = 1 - 54 = -53 \]

6. **Final result**: The division is complete. The quotient \( q(x) \) and remainder \( r(x) \) are:
\[ q(x) = x^2 - 4x - 18 \]
\[ r(x) = -53 \]

### Summary:
- Quotient \( q(x) = x^2 - 4x - 18 \)
- Remainder \( r(x) = -53 \)