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(x^5-4x^4+x^3-7x+1)/(x+2)

4 min read Jun 17, 2024
(x^5-4x^4+x^3-7x+1)/(x+2)

Dividing Polynomials: A Step-by-Step Guide for (x^5-4x^4+x^3-7x+1)/(x+2)

In this article, we will explore the process of dividing the polynomial (x^5-4x^4+x^3-7x+1) by (x+2) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is analogous to the long division of numbers. It involves systematically dividing the dividend polynomial by the divisor polynomial to obtain a quotient and remainder.

Steps for Dividing Polynomials

  1. Set up the division: Arrange the polynomials in descending order of exponents, ensuring all placeholders for missing terms are included.
        ________________________
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
  2. Divide the leading terms: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x). The result is x^4, which becomes the first term in the quotient.
        x^4 _______________
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
  3. Multiply the divisor by the quotient term: Multiply the divisor (x+2) by the quotient term (x^4) to get x^5 + 2x^4.
        x^4 _______________
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
       x^5 + 2x^4
  4. Subtract: Subtract the result from the dividend. This eliminates the leading term of the dividend.
        x^4 _______________
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
       x^5 + 2x^4
       ----------
           -6x^4 + x^3
  5. Bring down the next term: Bring down the next term of the dividend (-7x).
        x^4 _______________
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
       x^5 + 2x^4
       ----------
           -6x^4 + x^3 - 7x
  6. Repeat steps 2-5: Repeat steps 2-5 until the degree of the remainder is less than the degree of the divisor.
        x^4 - 6x^3 + 13x^2 - 26x + 45
x+2 | x^5 - 4x^4 + x^3 - 7x + 1
       x^5 + 2x^4
       ----------
           -6x^4 + x^3 
           -6x^4 - 12x^3
           ----------
                   13x^3 - 7x
                   13x^3 + 26x^2
                   ----------
                           -26x^2 - 7x
                           -26x^2 - 52x
                           ----------
                                   45x + 1
                                   45x + 90
                                   ----------
                                           -89

Result

Therefore, the division of (x^5-4x^4+x^3-7x+1) by (x+2) results in:

  • Quotient: x^4 - 6x^3 + 13x^2 - 26x + 45
  • Remainder: -89

We can express this result as:

(x^5-4x^4+x^3-7x+1)/(x+2) = x^4 - 6x^3 + 13x^2 - 26x + 45 - 89/(x+2)

This result can be verified by multiplying the quotient by the divisor and adding the remainder.

Polynomial long division is a fundamental process in algebra, allowing us to manipulate and analyze polynomials in a systematic manner.

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