(x^5-x^4+6x^3-2x^2+x-8)/(x-1)

Jasonbradley

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Jun 17, 2024

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Dividing Polynomials: (x^5-x^4+6x^3-2x^2+x-8)/(x-1)

This article will demonstrate how to divide the polynomial x^5-x^4+6x^3-2x^2+x-8 by x-1 using polynomial long division.

Polynomial Long Division

Polynomial long division is a method for dividing polynomials similar to long division with numbers. Here's how it works:

1. Set up the division: Write the dividend (x^5-x^4+6x^3-2x^2+x-8) inside the division symbol and the divisor (x-1) outside.

        ____________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
   

2. Divide 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 this term above the division symbol.

        x^4 __________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
   

3. Multiply and subtract: Multiply the divisor (x-1) by the term we just wrote (x^4), which gives us x^5 - x^4. Write this result below the dividend and subtract.

        x^4 __________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
           -(x^5 - x^4)
           ___________
                    6x^3 - 2x^2 + x - 8
   

4. Bring down the next term: Bring down the next term of the dividend (6x^3).

        x^4 __________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
           -(x^5 - x^4)
           ___________
                    6x^3 - 2x^2 + x - 8
   

5. Repeat steps 2-4: Repeat the process of dividing the leading term, multiplying, and subtracting.

        x^4 + 6x^2 __________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
           -(x^5 - x^4)
           ___________
                    6x^3 - 2x^2 + x - 8
                    -(6x^3 - 6x^2)
                    ___________
                            4x^2 + x - 8
   

6. Continue until the degree of the remainder is less than the degree of the divisor: Keep repeating the process until the degree of the remainder is less than the degree of the divisor (in this case, x-1).

        x^4 + 6x^2 + 4x __________
   x - 1 | x^5 - x^4 + 6x^3 - 2x^2 + x - 8 
           -(x^5 - x^4)
           ___________
                    6x^3 - 2x^2 + x - 8
                    -(6x^3 - 6x^2)
                    ___________
                            4x^2 + x - 8
                            -(4x^2 - 4x)
                            ___________
                                    5x - 8
   

7. Write the final result: The quotient is the expression above the division symbol: x^4 + 6x^2 + 4x. The remainder is the last expression obtained: 5x - 8.

Therefore, the result of dividing (x^5-x^4+6x^3-2x^2+x-8) by (x-1) is:

(x^5-x^4+6x^3-2x^2+x-8) / (x-1) = x^4 + 6x^2 + 4x + (5x - 8)/(x-1)

This result can be expressed as:

x^5-x^4+6x^3-2x^2+x-8 = (x-1)(x^4 + 6x^2 + 4x) + (5x - 8)

This shows how the original polynomial can be factored and written with a remainder.