Final Answer:
The result of dividing 3ˣ⁴ + 2ˣ³ - 7ˣ² - 12x - 2 by ˣ² - 2x + 1 is 3ˣ² + 8x - 10.
Step-by-step explanation:
To find the result of the division, we use polynomial long division. First, divide the leading term of the numerator, 3ˣ⁴, by the leading term of the denominator, ˣ², which gives 3ˣ². Multiply the entire denominator, ˣ² - 2x + 1, by this result and subtract it from the numerator. Repeat this process for each term in the numerator.
After performing the polynomial long division, the quotient is 3ˣ² + 8x - 10. This means that 3ˣ⁴ + 2ˣ³ - 7ˣ² - 12x - 2 can be expressed as the product of ˣ² - 2x + 1 and 3ˣ² + 8x - 10, plus a remainder of 0.
In conclusion, the division result is 3ˣ² + 8x - 10, and this indicates that the given polynomial is evenly divisible by ˣ² - 2x + 1. The process of polynomial long division ensures that the quotient, when multiplied by the divisor, yields the original polynomial, and the remainder is zero.