(x^3-5x^2-33x-35)/(x+3)

Jasonbradley

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Jun 17, 2024

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Solving the Polynomial Division: (x^3 - 5x^2 - 33x - 35) / (x + 3)

This article will guide you through the process of dividing the polynomial (x^3 - 5x^2 - 33x - 35) by (x + 3).

Polynomial Long Division

We will utilize the method of polynomial long division to solve this problem. Here's how it works:

1. Set up the division:

        ____________
   x + 3 | x^3 - 5x^2 - 33x - 35 
   

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

        x^2 _______
   x + 3 | x^3 - 5x^2 - 33x - 35 
   

3. Multiply the divisor by the quotient term: Multiply (x + 3) by x^2, resulting in x^3 + 3x^2. Write this product below the dividend.

        x^2 _______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
   

4. Subtract: Subtract the product from the dividend.

        x^2 _______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x
   

5. Bring down the next term: Bring down the next term of the dividend (-33x).

        x^2 _______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x - 35
   

6. Repeat the process: Now, divide the leading term of the new dividend (-8x^2) by the leading term of the divisor (x). This gives us -8x. Write this term above the dividend.

        x^2 - 8x ______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x - 35
   

7. Multiply and subtract: Multiply (x + 3) by -8x, resulting in -8x^2 - 24x. Subtract this from the current dividend.

        x^2 - 8x ______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x - 35
               -8x^2 - 24x 
               ---------
                      -9x - 35
   

8. Bring down the next term: Bring down the last term of the dividend (-35).

        x^2 - 8x ______
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x - 35
               -8x^2 - 24x 
               ---------
                      -9x - 35
   

9. Repeat again: Divide the leading term of the new dividend (-9x) by the leading term of the divisor (x). This gives us -9. Write this term above the dividend.

        x^2 - 8x - 9 ___
   x + 3 | x^3 - 5x^2 - 33x - 35 
           x^3 + 3x^2
           ---------
               -8x^2 - 33x - 35
               -8x^2 - 24x 
               ---------
                      -9x - 35
                      -9x - 27
                      ---------
                            -8 
   

10. Final subtraction: Multiply (x + 3) by -9, resulting in -9x - 27. Subtract this from the current dividend. The remainder is -8.

Result

Therefore, we can express the division as:

(x^3 - 5x^2 - 33x - 35) / (x + 3) = x^2 - 8x - 9 - 8/(x + 3)

This means the quotient is x^2 - 8x - 9, and the remainder is -8.