(5x^3-9x^2-6x+8)/(x+1)

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (5x^3 - 9x^2 - 6x + 8) / (x + 1)

This article will guide you through the process of dividing the polynomial (5x^3 - 9x^2 - 6x + 8) by the binomial (x + 1) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division method used for numbers. The goal is to find the quotient and remainder of the division.

Steps for Polynomial Long Division

1. Set up the division: Write the dividend (5x^3 - 9x^2 - 6x + 8) inside the division symbol and the divisor (x + 1) outside.

      __________
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
   

2. Divide the leading terms: Divide the leading term of the dividend (5x^3) by the leading term of the divisor (x), which gives 5x^2. Write this above the division symbol.

      5x^2 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
   

3. Multiply the quotient term by the divisor: Multiply 5x^2 by (x + 1) to get 5x^3 + 5x^2. Write this below the dividend.

      5x^2 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
   

4. Subtract: Subtract the result from the dividend.

      5x^2 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 
   

5. Bring down the next term: Bring down the next term from the dividend (-6x).

      5x^2 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
   

6. Repeat steps 2-5: Divide the leading term of the new dividend (-14x^2) by the leading term of the divisor (x), which gives -14x. Write this above the division symbol.

      5x^2 - 14x ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
                 -14x^2 - 14x
   

   Multiply -14x by (x + 1) and write the result below the new dividend. Subtract the result.

      5x^2 - 14x ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
                 -14x^2 - 14x
                 ---------
                       8x 
   

7. Bring down the next term: Bring down the next term from the dividend (+8).

      5x^2 - 14x ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
                 -14x^2 - 14x
                 ---------
                       8x + 8 
   

8. Repeat steps 2-5: Divide the leading term of the new dividend (8x) by the leading term of the divisor (x), which gives 8. Write this above the division symbol.

      5x^2 - 14x + 8 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
                 -14x^2 - 14x
                 ---------
                       8x + 8
                       8x + 8
   

   Multiply 8 by (x + 1) and write the result below the new dividend. Subtract the result.

      5x^2 - 14x + 8 ______
   x + 1 | 5x^3 - 9x^2 - 6x + 8 
           5x^3 + 5x^2
           ---------
                 -14x^2 - 6x 
                 -14x^2 - 14x
                 ---------
                       8x + 8
                       8x + 8
                       ---------
                           0
   

9. Result: The remainder is 0.

Therefore, (5x^3 - 9x^2 - 6x + 8) / (x + 1) = 5x^2 - 14x + 8.