(Score for Question 3:__ of 5 points) 3. Simplify the expression \( \left(x^{3}-5 x^{2}+7 x-12\right) \div(x-4) \) using long division. Show your work. Answer:

To simplify the expression \( (x^{3} - 5x^{2} + 7x - 12) \div (x - 4) \) using long division, we follow these steps: 1. Divide the leading term of the numerator \( x^3 \) by the leading term of the denominator \( x \) to get \( x^2 \). 2. Multiply \( x^2 \) by \( (x - 4) \) resulting in \( x^3 - 4x^2 \). 3. Subtract this result from the original polynomial: \[ (x^3 - 5x^2 + 7x - 12) - (x^3 - 4x^2) = -5x^2 + 4x^2 + 7x - 12 = -x^2 + 7x - 12. \] 4. Now divide the new leading term \( -x^2 \) by \( x \) to get \( -x \). 5. Multiply \( -x \) by \( (x - 4) \) yielding \( -x^2 + 4x \). 6. Subtract this result: \[ (-x^2 + 7x - 12) - (-x^2 + 4x) = 7x - 4x - 12 = 3x - 12. \] 7. Finally, divide \( 3x \) by \( x \) to get \( 3 \). 8. Multiply \( 3 \) by \( (x - 4) \) giving \( 3x - 12 \). 9. Subtracting results in: \[ (3x - 12) - (3x - 12) = 0. \] Thus, the result of the long division is \( x^2 - x + 3 \) with a remainder of \( 0 \). Final Answer: \( \frac{x^3 - 5x^2 + 7x - 12}{x - 4} = x^2 - x + 3. \)