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

Jasonbradley

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Jun 17, 2024

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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).