Polynomial Long Division: (6x^2  5x + 9) / (2x  1)
This article will guide you through the process of dividing the polynomial (6x^2  5x + 9) by (2x  1) using polynomial long division.
Steps:

Set up the division:
 Write the dividend (6x^2  5x + 9) inside the division symbol.
 Write the divisor (2x  1) outside the division symbol.

Divide the leading terms:
 Divide the leading term of the dividend (6x^2) by the leading term of the divisor (2x).
 This gives you 3x. Write this above the division symbol, aligning it with the x term.

Multiply the quotient by the divisor:
 Multiply the quotient (3x) by the entire divisor (2x  1).
 This gives you (6x^2  3x).

Subtract:
 Subtract the product (6x^2  3x) from the dividend (6x^2  5x + 9).
 This gives you (2x + 9).

Bring down the next term:
 Bring down the next term of the dividend (+9).

Repeat steps 25:
 Divide the new leading term (2x) by the leading term of the divisor (2x).
 This gives you 1. Write this next to the 3x above the division symbol.
 Multiply the new quotient (1) by the divisor (2x  1). This gives you (2x + 1).
 Subtract (2x + 1) from (2x + 9). This gives you 8.

The remainder:
 8 is the remainder, as it is a constant term and cannot be divided further by the divisor (2x  1).
Result:
The division of (6x^2  5x + 9) by (2x  1) results in:
(6x^2  5x + 9) / (2x  1) = 3x  1 + 8/(2x  1)
Therefore, the quotient is 3x  1 and the remainder is 8.
Conclusion:
Polynomial long division is a fundamental technique for dividing polynomials. By following the steps outlined above, you can successfully divide any polynomial by another polynomial. It is a crucial tool in various mathematical applications, including simplifying expressions, solving equations, and understanding the behavior of functions.