Find the quotient and the remainder when `x^4 -6x^3 +3x^2 + 26 x - 24` is divided by ` x-4`

Answer

To find the quotient and remainder when dividing the polynomial \( P(x) = x^4 - 6x^3 + 3x^2 + 26x - 24 \) by \( D(x) = x - 4 \), we can use polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: Write the dividend \( P(x) \) and the divisor \( D(x) \):
\[ \text{Dividend: } P(x) = x^4 - 6x^3 + 3x^2 + 26x - 24 \]
\[ \text{Divisor: } D(x) = x - 4 \]

2. **Divide the leading term**: Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{x^4}{x} = x^3 \]
This is the first term of the quotient.

3. **Multiply and subtract**: Multiply \( x^3 \) by the entire divisor \( D(x) \) and subtract from \( P(x) \):
\[ x^3 \cdot (x - 4) = x^4 - 4x^3 \]
Subtract this from \( P(x) \):
\[ (x^4 - 6x^3 + 3x^2 + 26x - 24) - (x^4 - 4x^3) = -2x^3 + 3x^2 + 26x - 24 \]

4. **Repeat the process**: Now, repeat the process with the new polynomial \( -2x^3 + 3x^2 + 26x - 24 \).
Divide the leading term:
\[ \frac{-2x^3}{x} = -2x^2 \]
Multiply and subtract:
\[ -2x^2 \cdot (x - 4) = -2x^3 + 8x^2 \]
Subtract:
\[ (-2x^3 + 3x^2 + 26x - 24) - (-2x^3 + 8x^2) = -5x^2 + 26x - 24 \]

5. **Continue the division**: Divide the leading term:
\[ \frac{-5x^2}{x} = -5x \]
Multiply and subtract:
\[ -5x \cdot (x - 4) = -5x^2 + 20x \]
Subtract:
\[ (-5x^2 + 26x - 24) - (-5x^2 + 20x) = 6x - 24 \]

6. **Final division step**: Divide the leading term:
\[ \frac{6x}{x} = 6 \]
Multiply and subtract:
\[ 6 \cdot (x - 4) = 6x - 24 \]
Subtract:
\[ (6x - 24) - (6x - 24) = 0 \]

7. **Conclusion**: The division is complete. The quotient is:
\[ Q(x) = x^3 - 2x^2 - 5x + 6 \]
And the remainder is:
\[ R = 0 \]

### Final Answer:
- Quotient: \( x^3 - 2x^2 - 5x + 6 \)
- Remainder: \( 0 \)