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(3x^3-x^2-7x+6)/(x+2) Long Division

5 min read Jun 16, 2024
(3x^3-x^2-7x+6)/(x+2) Long Division

Long Division of Polynomials: (3x^3 - x^2 - 7x + 6) / (x + 2)

Long division of polynomials is a method used to divide a polynomial by another polynomial of lesser or equal degree. Here's how to perform the long division of (3x^3 - x^2 - 7x + 6) by (x + 2):

1. Setting Up the Division:

        ___________
x + 2 | 3x^3 - x^2 - 7x + 6

2. Dividing the Leading Terms:

  • Focus on the leading terms of both polynomials: 3x^3 and x.
  • Divide the leading term of the dividend (3x^3) by the leading term of the divisor (x) : 3x^3 / x = 3x^2.
  • Write 3x^2 above the line in the quotient.
        3x^2 ______
x + 2 | 3x^3 - x^2 - 7x + 6

3. Multiplying the Quotient by the Divisor:

  • Multiply the quotient term (3x^2) by the entire divisor (x + 2): 3x^2 * (x + 2) = 3x^3 + 6x^2.
  • Write the result below the dividend, aligning terms with their corresponding degrees.
        3x^2 ______
x + 2 | 3x^3 - x^2 - 7x + 6 
        -(3x^3 + 6x^2)

4. Subtracting the Result:

  • Subtract the result (3x^3 + 6x^2) from the dividend. Remember to change the signs when subtracting.
        3x^2 ______
x + 2 | 3x^3 - x^2 - 7x + 6 
        -(3x^3 + 6x^2) 
        ----------------
             -7x^2 - 7x

5. Bringing Down the Next Term:

  • Bring down the next term of the dividend (-7x) next to the result.
        3x^2 ______
x + 2 | 3x^3 - x^2 - 7x + 6 
        -(3x^3 + 6x^2) 
        ----------------
             -7x^2 - 7x

6. Repeating Steps 2-5:

  • Repeat the process from step 2:
    • Divide the leading term of the new dividend (-7x^2) by the leading term of the divisor (x): -7x^2 / x = -7x.
    • Write -7x next to 3x^2 in the quotient.
    • Multiply -7x by the divisor (x + 2): -7x * (x + 2) = -7x^2 - 14x.
    • Write the result below the new dividend.
    • Subtract the result, changing signs:
        3x^2 - 7x _____
x + 2 | 3x^3 - x^2 - 7x + 6 
        -(3x^3 + 6x^2) 
        ----------------
             -7x^2 - 7x 
             -(-7x^2 - 14x)
             --------------
                     7x + 6

7. Final Steps:

  • Bring down the last term of the dividend (6).
  • Divide the leading term of the new dividend (7x) by the leading term of the divisor (x): 7x / x = 7.
  • Write 7 in the quotient.
  • Multiply 7 by the divisor (x + 2): 7 * (x + 2) = 7x + 14.
  • Write the result below the new dividend.
  • Subtract the result:
        3x^2 - 7x + 7 ___
x + 2 | 3x^3 - x^2 - 7x + 6 
        -(3x^3 + 6x^2) 
        ----------------
             -7x^2 - 7x 
             -(-7x^2 - 14x)
             --------------
                     7x + 6 
                     -(7x + 14)
                     ---------
                           -8

Result:

The result of the long division is:

(3x^3 - x^2 - 7x + 6) / (x + 2) = 3x^2 - 7x + 7 - 8/(x + 2)

Therefore, the quotient is 3x^2 - 7x + 7 and the remainder is -8.

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