(2x^5+x^4-15x^3-2x^2+10x-24)/(x^2-x-4)

Jasonbradley

7 min read

·

Jun 16, 2024

38

Share

Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 by x^2 - x - 4.

The Long Division Method

We will be using the long division method for polynomials. This method is similar to the long division you learned for numbers.

1. Set up the division. Write the dividend (2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24) inside the division symbol and the divisor (x^2 - x - 4) outside.

       ____________
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
   

2. Divide the leading terms. Focus on the leading term of the divisor (x^2) and the leading term of the dividend (2x^5). What do you need to multiply x^2 by to get 2x^5? The answer is 2x^3.

       2x^3 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
   

3. Multiply the quotient by the divisor. Multiply 2x^3 by (x^2 - x - 4) and write the result below the dividend.

       2x^3 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
   

4. Subtract. Change the signs of the terms in the second row and add.

       2x^3 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3  
   

5. Bring down the next term. Bring down the next term of the dividend (-2x^2).

       2x^3 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3 - 2x^2 
   

6. Repeat steps 2-5. Now, focus on the leading term of the new dividend (3x^4) and the leading term of the divisor (x^2). What do you need to multiply x^2 by to get 3x^4? The answer is 3x^2.

       2x^3 + 3x^2 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3 - 2x^2 
                  3x^4 - 3x^3 - 12x^2
   

7. Subtract and bring down. Change the signs of the second row, add, and bring down the next term (10x).

       2x^3 + 3x^2 _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3 - 2x^2 
                  3x^4 - 3x^3 - 12x^2
                  ------------------
                        -4x^3 + 10x^2 + 10x 
   

8. Repeat steps 2-5. Continue this process until the degree of the dividend is less than the degree of the divisor.

       2x^3 + 3x^2 - 4x _______
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3 - 2x^2 
                  3x^4 - 3x^3 - 12x^2
                  ------------------
                        -4x^3 + 10x^2 + 10x 
                        -4x^3 + 4x^2 + 16x
                        ------------------
                                6x^2 - 6x - 24
   

9. Final step. Focus on the leading term of the new dividend (6x^2) and the leading term of the divisor (x^2). What do you need to multiply x^2 by to get 6x^2? The answer is 6.

       2x^3 + 3x^2 - 4x + 6 
   x^2-x-4 | 2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24 
            2x^5 - 2x^4 - 8x^3 
            ------------------
                  3x^4 - 7x^3 - 2x^2 
                  3x^4 - 3x^3 - 12x^2
                  ------------------
                        -4x^3 + 10x^2 + 10x 
                        -4x^3 + 4x^2 + 16x
                        ------------------
                                6x^2 - 6x - 24 
                                6x^2 - 6x - 24
                                ----------------
                                        0 
   

The Result

The result of the division is:

(2x^5 + x^4 - 15x^3 - 2x^2 + 10x - 24) / (x^2 - x - 4) = 2x^3 + 3x^2 - 4x + 6

Important Note:

This method can be applied to any division of polynomials. The key is to remember to focus on the leading terms of the divisor and the dividend at each step.