Final answer:
The polynomial long division of (7x³ + x² + x) / (x²+1) results in a quotient of 7x - 6 with a remainder of x + 6. By following the steps of dividing, multiplying, and subtracting until the new remainder has a lower degree than the divisor, we reach the conclusion of the division process.
Step-by-step explanation:
To divide (7x³ + x² + x) / (x²+1) using polynomial long division, we must follow a process similar to that of long division for numerical values. Below is a step-by-step explanation of how to perform polynomial long division with this specific example:
1. Divide the first term of the dividend (7x³) by the first term of the divisor (x²), which gives us 7x. Write this above the division bar.
2. Multiply the entire divisor (x²+1) by this quotient (7x), resulting in (7x³ + 7x).
3. Subtract this from the original dividend, leaving you with -6x² + x.
4. Repeat these steps by dividing the new first term (-6x²) by the first term of the divisor (x²), giving -6. Place this term above the division bar next to 7x.
5. Multiply the divisor by -6, yielding (-6x² - 6).
6. Subtract again to get a new remainder of x + 6.
7. Finally, as x is of a lower degree than the first term of the divisor (x²), this is the final remainder, and the process is complete.
The result of the division is the quotient 7x - 6 with a remainder of x + 6. Therefore, the solution to the long division problem is (7x³ + x² + x) / (x²+1) equals 7x - 6 with a remainder of x + 6.