(2x^3+7x^2-6x-8) Divided By (x+4)

4 min read Jun 16, 2024
(2x^3+7x^2-6x-8) Divided By (x+4)

Dividing Polynomials: (2x^3 + 7x^2 - 6x - 8) ÷ (x + 4)

This article will walk through the process of dividing the polynomial 2x³ + 7x² - 6x - 8 by the binomial x + 4. We'll use the method of polynomial long division.

Step 1: Set up the division problem

           _________
x + 4 | 2x³ + 7x² - 6x - 8 

Step 2: Divide the leading terms

  • Focus on the leading terms of the divisor (x) and the dividend (2x³).
  • Ask: What do I multiply x by to get 2x³? The answer is 2x².
  • Write 2x² above the division bar, aligned with the x² term.
           2x²        
x + 4 | 2x³ + 7x² - 6x - 8 

Step 3: Multiply and subtract

  • Multiply the entire divisor (x + 4) by 2x². This gives us 2x³ + 8x².
  • Write this result below the dividend.
  • Subtract the two expressions.
           2x²        
x + 4 | 2x³ + 7x² - 6x - 8 
        -(2x³ + 8x²)
          _________
               -x² - 6x

Step 4: Bring down the next term

  • Bring down the next term of the dividend (-6x).
           2x²        
x + 4 | 2x³ + 7x² - 6x - 8 
        -(2x³ + 8x²)
          _________
               -x² - 6x

Step 5: Repeat steps 2-4

  • Focus on the new leading term of the dividend (-x²) and the leading term of the divisor (x).
  • Ask: What do I multiply x by to get -x²? The answer is -x.
  • Write -x above the division bar, aligned with the x term.
           2x² - x      
x + 4 | 2x³ + 7x² - 6x - 8 
        -(2x³ + 8x²)
          _________
               -x² - 6x
               -(-x² - 4x)
               _________
                      -2x - 8

Step 6: Bring down the next term and repeat

  • Bring down the next term of the dividend (-8).
  • Focus on the new leading term of the dividend (-2x) and the leading term of the divisor (x).
  • Ask: What do I multiply x by to get -2x? The answer is -2.
  • Write -2 above the division bar, aligned with the constant term.
           2x² - x - 2    
x + 4 | 2x³ + 7x² - 6x - 8 
        -(2x³ + 8x²)
          _________
               -x² - 6x
               -(-x² - 4x)
               _________
                      -2x - 8
                      -(-2x - 8)
                      _________
                              0

Conclusion

The division is complete, and we have a remainder of 0. Therefore, the result of dividing 2x³ + 7x² - 6x - 8 by x + 4 is 2x² - x - 2.

In other words:

2x³ + 7x² - 6x - 8 = (x + 4)(2x² - x - 2)

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