Answer :
Final Answer:
The result of the division (-3x⁵ - 22x⁴ - 13x³ + 39x² + 14x - 6) / (x³ + 6x² - 3x - 5) is -3x² - 4x + 1.
Explanation:
To perform the division of the polynomial (-3x⁵ - 22x⁴ - 13x³ + 39x² + 14x - 6) by (x³ + 6x² - 3x - 5), we use polynomial long division. This process is similar to long division with numbers, but we divide each term of the dividend by the first term of the divisor.
We start by dividing the highest degree term of the dividend, which is -3x⁵, by the highest degree term of the divisor, which is x³. This results in -3x². We then multiply the entire divisor (x³ + 6x² - 3x - 5) by -3x² and subtract this product from the dividend:
(-3x⁵ - 22x⁴ - 13x³ + 39x² + 14x - 6) - (-3x² * (x³ + 6x² - 3x - 5))
After simplifying the above expression, we get:
-4x⁴ - 10x³ + 45x² + 14x - 6
Now, we repeat the process by dividing the highest degree term of this new expression, which is -4x⁴, by the highest degree term of the divisor, which is x³. This results in -4x. We continue the process until all terms have been divided. Ultimately, the result of the division is -3x² - 4x + 1.
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