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(x^5-3x^4-7x+18)/(x-3)

4 min read Jun 17, 2024
(x^5-3x^4-7x+18)/(x-3)

Dividing Polynomials: (x^5 - 3x^4 - 7x + 18) / (x - 3)

This article explores the process of dividing the polynomial x^5 - 3x^4 - 7x + 18 by the binomial x - 3. We'll use polynomial long division to solve this problem.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials that mirrors the long division process used for numbers. Here's a breakdown of the steps:

  1. Set up the division:

    • Write the dividend (x^5 - 3x^4 - 7x + 18) inside the division symbol.
    • Write the divisor (x - 3) outside the division symbol.

  2. Focus on the leading terms:

    • Divide the leading term of the dividend (x^5) by the leading term of the divisor (x). This gives us x^4.
    • Write x^4 above the dividend, aligning it with the x^4 term.

  3. Multiply the divisor by the quotient:

    • Multiply (x - 3) by x^4 to get x^5 - 3x^4.
    • Write this result below the dividend.

  4. Subtract:

    • Subtract the result from step 3 from the dividend. Note that the x^5 and x^4 terms will cancel out, leaving us with -7x + 18.

  5. Bring down the next term:

    • Bring down the next term from the dividend (-7x) to form the new leading term.

  6. Repeat steps 2-5:

    • Divide the new leading term (-7x) by the divisor's leading term (x), giving us -7.
    • Write -7 above the dividend, aligning it with the constant term.
    • Multiply (x - 3) by -7 to get -7x + 21.
    • Subtract this result from the current dividend.
    • Bring down the remaining term (18).

  7. Final Step:

    • Divide the new leading term (18) by the divisor's leading term (x). This gives us 18/x.
    • Since the degree of the numerator (0) is less than the degree of the denominator (1), we cannot continue dividing. This becomes our remainder.

Solution

Here's the complete solution using polynomial long division:

        x^4     - 7      + 18/x 
    _______________________
x - 3 | x^5 - 3x^4     - 7x + 18
        -(x^5 - 3x^4)
        ------------------
              0          - 7x + 18
              -  (-7x + 21)
              ------------------
                        -3

Therefore, the result of dividing (x^5 - 3x^4 - 7x + 18) by (x - 3) is:

x^4 - 7 + 18/x - 3/x-3

Conclusion

Polynomial long division is a fundamental tool for working with polynomials. It allows us to divide polynomials and express the result as a quotient and a remainder. The process, while seemingly complex, follows a straightforward pattern, making it accessible with practice. Understanding this process is crucial for tackling more advanced algebraic concepts and problem-solving in mathematics.

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