(2x4 + 4x3 + 2x2 + 8x + 8) ÷ (x + 2)

Jasonbradley

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Jun 16, 2024

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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.