(2x^3+8x^2-6x+10)/(x-2)

Jasonbradley

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Jun 16, 2024

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Polynomial Division: (2x^3 + 8x^2 - 6x + 10) / (x - 2)

In this article, we will be performing polynomial division on the expression (2x^3 + 8x^2 - 6x + 10) / (x - 2).

Long Division Method

The most common method for polynomial division is long division. Let's break down the steps:

1. Set up the problem: Write the dividend (2x^3 + 8x^2 - 6x + 10) inside the division symbol and the divisor (x - 2) outside.

      ___________
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
   

2. Divide the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x). This gives us 2x^2. Write this above the division symbol.

      2x^2 _______
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
   

3. Multiply and subtract: Multiply the quotient (2x^2) by the divisor (x - 2). This gives us 2x^3 - 4x^2. Write this result below the dividend and subtract.

      2x^2 _______
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
          -(2x^3 - 4x^2)
          --------------
                 12x^2 
   

4. Bring down the next term: Bring down the next term of the dividend (-6x).

      2x^2 _______
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
          -(2x^3 - 4x^2)
          --------------
                 12x^2 - 6x 
   

5. Repeat steps 2-4: Divide the new leading term (12x^2) by the leading term of the divisor (x). This gives us 12x. Write this above the division symbol. Multiply 12x by the divisor (x - 2) and subtract.

      2x^2 + 12x _______
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
          -(2x^3 - 4x^2)
          --------------
                 12x^2 - 6x 
                 -(12x^2 - 24x)
                 -------------
                           18x
   

6. Repeat again: Bring down the next term (10) and repeat the process. Divide 18x by x, which gives us 18. Multiply 18 by (x - 2) and subtract.

      2x^2 + 12x + 18 ______
   x - 2 | 2x^3 + 8x^2 - 6x + 10 
          -(2x^3 - 4x^2)
          --------------
                 12x^2 - 6x 
                 -(12x^2 - 24x)
                 -------------
                           18x + 10
                           -(18x - 36)
                           ------------
                                 46 
   

7. Result: The quotient is 2x^2 + 12x + 18 and the remainder is 46. We can write this result as:

   (2x^3 + 8x^2 - 6x + 10) / (x - 2) = 2x^2 + 12x + 18 + 46/(x - 2)

Conclusion

By performing long division, we found that the quotient of (2x^3 + 8x^2 - 6x + 10) divided by (x - 2) is 2x^2 + 12x + 18 with a remainder of 46.