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(x^5+6x^4-3x^2-22x-29)/(x+6)

4 min read Jun 17, 2024
(x^5+6x^4-3x^2-22x-29)/(x+6)

Dividing Polynomials: (x^5 + 6x^4 - 3x^2 - 22x - 29) / (x + 6)

This article will guide you through the process of dividing the polynomial (x^5 + 6x^4 - 3x^2 - 22x - 29) by (x + 6) using polynomial long division.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division process used with numbers. Here's how it works:

  1. Set up the division: Write the dividend (x^5 + 6x^4 - 3x^2 - 22x - 29) inside the division symbol and the divisor (x + 6) outside.

  2. Divide the leading terms: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x). This gives you x^4. Write this term above the dividend.

  3. Multiply the divisor: Multiply the quotient term (x^4) by the entire divisor (x + 6). This gives you x^5 + 6x^4.

  4. Subtract: Subtract the result from the dividend. This leaves you with 0.

  5. Bring down the next term: Bring down the next term from the dividend (-3x^2).

  6. Repeat steps 2-5: Now divide the leading term of the new dividend (-3x^2) by the leading term of the divisor (x), which gives you -3x. Write this term above the dividend. Multiply the divisor by -3x, subtract the result, and bring down the next term (-22x).

  7. Continue the process: Continue this process until you have no more terms to bring down.

Here's the long division process step-by-step:

             x^4  - 3x      + 17 
       _______________________
x + 6 | x^5 + 6x^4 - 3x^2 - 22x - 29
        -(x^5 + 6x^4) 
        _______________________
                 - 3x^2 - 22x
                 -(-3x^2 - 18x)
                 _______________________
                          - 4x - 29
                          -(-4x - 24)
                          _______________________
                                  - 5

Result

The result of dividing (x^5 + 6x^4 - 3x^2 - 22x - 29) by (x + 6) is:

x^4 - 3x + 17 with a remainder of -5.

This can be written as:

(x^5 + 6x^4 - 3x^2 - 22x - 29) / (x + 6) = x^4 - 3x + 17 - 5/(x + 6)

Conclusion

Polynomial long division allows you to divide complex polynomials by simpler ones, finding the quotient and remainder. This is a useful technique in algebra and calculus for simplifying expressions and solving equations.

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