Dividing Polynomials: (7x^3 + x^2 + x) / (x^2 + 1)
This article explores the division of the polynomial (7x^3 + x^2 + x) by (x^2 + 1) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is analogous to long division with numbers. The goal is to find the quotient and remainder when dividing one polynomial by another.
Steps for Polynomial Long Division:

Set up the division: Write the dividend (7x^3 + x^2 + x) inside the division symbol and the divisor (x^2 + 1) outside.

Divide the leading terms: Divide the leading term of the dividend (7x^3) by the leading term of the divisor (x^2), resulting in 7x. Write this term above the division symbol.

Multiply the divisor by the quotient term: Multiply the divisor (x^2 + 1) by 7x, resulting in (7x^3 + 7x). Write this result below the dividend.

Subtract: Subtract the result from the dividend. In this case, (7x^3 + x^2 + x)  (7x^3 + 7x) = x^2  6x.

Bring down the next term: Bring down the next term of the dividend (x) to get x^2  6x + x.

Repeat steps 25: Now divide the leading term of the new dividend (x^2) by the leading term of the divisor (x^2), resulting in 1. Write this term above the division symbol next to 7x.

Multiply and subtract: Multiply (x^2 + 1) by 1, resulting in (x^2 + 1). Subtract this from the new dividend (x^2  6x + x), getting 6x.

Repeat again: Divide the leading term of the new dividend (6x) by the leading term of the divisor (x^2). Since the degree of the dividend (6x) is less than the degree of the divisor (x^2), the division stops.
The Result
The result of the division is:
(7x^3 + x^2 + x) / (x^2 + 1) = 7x + 1 + (6x)/(x^2 + 1)
Therefore, the quotient is (7x + 1) and the remainder is (6x).
This can be expressed as:
(7x^3 + x^2 + x) = (x^2 + 1)(7x + 1)  6x
Conclusion
Polynomial long division provides a systematic way to divide polynomials. The process involves repeatedly dividing the leading terms, multiplying, subtracting, and bringing down the next term until the degree of the dividend is less than the degree of the divisor.