(3x4 – 4x2 + 8x – 1) ÷ (x – 2)

4 min read Jun 16, 2024

Polynomial Long Division: (3x⁴ – 4x² + 8x – 1) ÷ (x – 2)

This article will demonstrate how to perform polynomial long division for the expression (3x⁴ – 4x² + 8x – 1) ÷ (x – 2).

Step 1: Setting up the Division

Begin by setting up the division problem like a traditional long division problem:

          ____________
x - 2 | 3x⁴ - 4x² + 8x - 1

Step 2: Divide the Leading Terms

Divide the leading term of the dividend (3x⁴) by the leading term of the divisor (x). This gives us 3x³.
Write this result (3x³) above the dividend.

          3x³ _________
x - 2 | 3x⁴ - 4x² + 8x - 1

Step 3: Multiply and Subtract

Multiply the divisor (x - 2) by the term we just wrote (3x³): (x - 2) * (3x³) = 3x⁴ - 6x³.
Write this result below the dividend and subtract.

          3x³ _________
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x²

Step 4: Bring Down the Next Term

Bring down the next term from the dividend (8x):

          3x³ _________
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x

Step 5: Repeat Steps 2-4

Divide the new leading term (2x³) by the leading term of the divisor (x): 2x³ / x = 2x².
Write this result above the dividend:

          3x³ + 2x² _________
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x

Multiply the divisor by the new term (2x²): (x - 2) * (2x²) = 2x³ - 4x².
Write this result below the previous line and subtract:

          3x³ + 2x² _________
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x
                  -(2x³ - 4x²)
                  _________
                            8x - 1

Step 6: Repeat Again

Bring down the next term from the dividend (-1):

          3x³ + 2x² _________
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x
                  -(2x³ - 4x²)
                  _________
                            8x - 1

Divide the new leading term (8x) by the leading term of the divisor (x): 8x / x = 8.
Write this result above the dividend:

          3x³ + 2x² + 8 _______
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x
                  -(2x³ - 4x²)
                  _________
                            8x - 1

Multiply the divisor by the new term (8): (x - 2) * (8) = 8x - 16.
Write this result below the previous line and subtract:

          3x³ + 2x² + 8 _______
x - 2 | 3x⁴ - 4x² + 8x - 1 
          -(3x⁴ - 6x³)
          _________
                  2x³ - 4x² + 8x
                  -(2x³ - 4x²)
                  _________
                            8x - 1
                            -(8x - 16)
                            _________
                                    15

Step 7: The Result

We have reached a point where the degree of the remainder (15) is less than the degree of the divisor (x - 2). This means we have completed the division.

Therefore, the result of (3x⁴ – 4x² + 8x – 1) ÷ (x – 2) is 3x³ + 2x² + 8 + 15/(x - 2).