(2x4 + 4x3 + 2x2 + 8x + 8) ÷ (x + 2)

5 min read Jun 16, 2024
(2x4 + 4x3 + 2x2 + 8x + 8) ÷ (x + 2)

Polynomial Long Division: (2x⁴ + 4x³ + 2x² + 8x + 8) ÷ (x + 2)

This article will demonstrate how to perform polynomial long division to find the quotient and remainder of the expression (2x⁴ + 4x³ + 2x² + 8x + 8) ÷ (x + 2).

Polynomial Long Division Steps

  1. Set up the division: Write the dividend (2x⁴ + 4x³ + 2x² + 8x + 8) inside the division symbol and the divisor (x + 2) outside.

        ____________
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
    
  2. Divide the leading terms: Divide the leading term of the dividend (2x⁴) by the leading term of the divisor (x), which gives 2x³. Write this result above the division symbol.

        2x³_________
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
    
  3. Multiply the divisor by the result: Multiply the divisor (x + 2) by the result (2x³), which gives 2x⁴ + 4x³. Write this result below the dividend.

        2x³_________
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
    
  4. Subtract: Subtract the result from the dividend.

        2x³_________
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
    
  5. Bring down the next term: Bring down the next term of the dividend (8x).

        2x³_________
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
    
  6. Repeat steps 2-5: Now we divide the leading term of the new dividend (2x²) by the leading term of the divisor (x), which gives 2x. Write this result above the division symbol.

        2x³ + 2x______
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
                 2x² + 4x
    

    Multiply the divisor (x + 2) by the result (2x) to get 2x² + 4x. Subtract this from the previous result.

        2x³ + 2x______
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
                 2x² + 4x
                 _______
                       4x + 8
    

    Bring down the next term (8).

        2x³ + 2x______
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
                 2x² + 4x
                 _______
                       4x + 8
    
  7. Final step: Divide the leading term of the new dividend (4x) by the leading term of the divisor (x), which gives 4. Write this above the division symbol.

        2x³ + 2x + 4___
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
                 2x² + 4x
                 _______
                       4x + 8
                       4x + 8
    

    Multiply the divisor (x + 2) by the result (4), which gives 4x + 8. Subtract this from the previous result.

        2x³ + 2x + 4___
    x + 2 | 2x⁴ + 4x³ + 2x² + 8x + 8 
           2x⁴ + 4x³
           _________
                 2x² + 8x + 8 
                 2x² + 4x
                 _______
                       4x + 8
                       4x + 8
                       _____
                           0
    

Result

Therefore, (2x⁴ + 4x³ + 2x² + 8x + 8) ÷ (x + 2) = 2x³ + 2x + 4, with a remainder of 0.