Final Answer:
The result of dividing x³ - 5x² - 39x - 12 ÷ x + 4 is x² - 9x + 3.
Step-by-step explanation:
To divide the given polynomial by x + 4, we can use long division. Start by dividing the first term of the numerator x³ by the first term of the denominator x, which gives x². Multiply x + 4 by x² to get x³ + 4x², and subtract this from the original polynomial. The remainder is -9x² - 39x - 12.
Repeat the process by dividing the leading term of the remainder -9x² by the leading term of the divisor x, resulting in -9x. Multiply x + 4 by -9xand subtract to get a new remainder of 3x - 12. Lastly, divide the leading term of this remainder 3x by the leading term of the divisor x, obtaining 3. Multiply x + 4 by 3 and subtract to get a final remainder of 0. Thus, the quotient is x² - 9x + 3.
This process follows the standard long division method, breaking down the division into steps. Each step ensures that the terms are appropriately subtracted and the next term of the quotient is determined. The final result is a polynomial without any remainder, indicating a clean division. The obtained quotient x² - 9x + 3 is the solution to the given division problem.