(x^3-4x-10)/(x^2-x-6) Long Division

3 min read Jun 17, 2024
(x^3-4x-10)/(x^2-x-6) Long Division

Long Division of Polynomials: (x^3 - 4x - 10) / (x^2 - x - 6)

Long division is a fundamental technique in algebra used to divide polynomials. Let's explore how to divide the polynomial (x^3 - 4x - 10) by (x^2 - x - 6).

Step-by-Step Solution

  1. Set up the division problem:

         _______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
    
    • We add a placeholder term (0x^2) in the dividend to maintain order.
  2. Divide the leading terms:

    • (x^3) / (x^2) = x
    • Write 'x' above the x term in the quotient.
         x ______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
    
  3. Multiply the divisor by the quotient term:

    • x * (x^2 - x - 6) = x^3 - x^2 - 6x
         x ______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
            -(x^3 - x^2 - 6x)
            ----------------
    
  4. Subtract the result:

    • (x^3 + 0x^2 - 4x) - (x^3 - x^2 - 6x) = x^2 + 2x
         x ______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
            -(x^3 - x^2 - 6x)
            ----------------
                 x^2 + 2x - 10
    
  5. Bring down the next term:

         x ______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
            -(x^3 - x^2 - 6x)
            ----------------
                 x^2 + 2x - 10
    
  6. Repeat steps 2-5 with the new polynomial:

    • (x^2) / (x^2) = 1
    • Write '1' above the constant term in the quotient.
         x + 1 ______
    x^2-x-6 | x^3 + 0x^2 - 4x - 10
            -(x^3 - x^2 - 6x)
            ----------------
                 x^2 + 2x - 10
                 -(x^2 - x - 6)
                 ----------------
                         3x - 4
    
  7. The degree of the remainder (3x - 4) is less than the degree of the divisor (x^2 - x - 6), so we stop here.

Result

Therefore, the result of the long division is:

(x^3 - 4x - 10) / (x^2 - x - 6) = x + 1 + (3x - 4)/(x^2 - x - 6)

This means the quotient is (x + 1) and the remainder is (3x - 4).