(2x^5+3x^4+8x^3+8x^2+18x+9)/(x^2+2x+3)

Jasonbradley

4 min read

·

Jun 16, 2024

38

Share

Dividing Polynomials: (2x^5+3x^4+8x^3+8x^2+18x+9) / (x^2+2x+3)

This article will demonstrate the process of dividing the polynomial (2x^5+3x^4+8x^3+8x^2+18x+9) by (x^2+2x+3) using polynomial long division.

Polynomial Long Division

Polynomial long division is similar to regular long division, but instead of dividing numbers, we are dividing polynomials. Here's how it works:

1. Set up the division problem: Write the dividend (2x^5+3x^4+8x^3+8x^2+18x+9) inside the division symbol and the divisor (x^2+2x+3) outside.

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

3. Multiply the divisor: Multiply the divisor (x^2+2x+3) by the term we just found (2x^3). This gives us 2x^5 + 4x^4 + 6x^3. Write this result below the dividend.

4. Subtract: Subtract the result from step 3 from the dividend. This gives us -x^4 + 2x^3 + 8x^2 + 18x + 9.

5. Bring down the next term: Bring down the next term of the dividend (18x).

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

7. Continue the process: Repeat steps 2-5 until the degree of the remaining polynomial is less than the degree of the divisor.

Performing the Division

Here is the detailed division process:

             2x^3 - x^2 + 4x + 2 
       _________________________
x^2+2x+3 | 2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9
             -(2x^5 + 4x^4 + 6x^3)
             -----------------------
                  -x^4 + 2x^3 + 8x^2 + 18x
                  -(-x^4 - 2x^3 - 3x^2)
                  -----------------------
                         4x^3 + 11x^2 + 18x
                         -(4x^3 + 8x^2 + 12x)
                         -----------------------
                                3x^2 + 6x + 9
                                -(3x^2 + 6x + 9)
                                -----------------
                                           0 

Result

Therefore, the result of dividing (2x^5+3x^4+8x^3+8x^2+18x+9) by (x^2+2x+3) is 2x^3 - x^2 + 4x + 2.

Conclusion

Polynomial long division is a useful tool for dividing polynomials. By following the steps outlined above, you can successfully divide any polynomial by another polynomial of a lower or equal degree.