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(6x^2-5x+9)/(2x-1)

3 min read Jun 16, 2024
(6x^2-5x+9)/(2x-1)

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:

  1. Set up the division:

    • Write the dividend (6x^2 - 5x + 9) inside the division symbol.
    • Write the divisor (2x - 1) outside the division symbol.

  2. 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.

  3. Multiply the quotient by the divisor:

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

  4. Subtract:

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

  5. Bring down the next term:

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

  6. Repeat steps 2-5:

    • 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.

  7. 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.

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