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(x^4-17x^2+4x-2)/(x^2+4x-1)

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

Dividing Polynomials: (x^4 - 17x^2 + 4x - 2) / (x^2 + 4x - 1)

This article will guide you through the process of dividing the polynomial (x^4 - 17x^2 + 4x - 2) by (x^2 + 4x - 1). We will use polynomial long division to achieve this.

Polynomial Long Division

Polynomial long division works similarly to regular long division, but instead of digits, we work with terms containing variables. Let's break down the steps:

  1. Set up the division:

         ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2
    • Notice we add a 0x^3 term for clarity in our setup.

  2. Divide the leading terms:

    • Divide the leading term of the divisor (x^2) into the leading term of the dividend (x^4). This gives us x^2.
    • Write x^2 above the dividend.
         x^2       
     ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2

  3. Multiply and subtract:

    • Multiply the quotient term (x^2) by the divisor (x^2 + 4x - 1): x^2 * (x^2 + 4x - 1) = x^4 + 4x^3 - x^2.
    • Subtract this result from the dividend.
         x^2       
     ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2 
           -(x^4 + 4x^3 - x^2)
           --------------------
               -4x^3 - 16x^2 + 4x

  4. Repeat the process:

    • Bring down the next term from the dividend (-2).
    • Divide the new leading term (-4x^3) by the leading term of the divisor (x^2). This gives us -4x.
    • Write -4x next to the x^2 in the quotient.
         x^2 - 4x       
     ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2 
           -(x^4 + 4x^3 - x^2)
           --------------------
               -4x^3 - 16x^2 + 4x

  5. Continue multiplying and subtracting:

    • Multiply the new quotient term (-4x) by the divisor: -4x * (x^2 + 4x - 1) = -4x^3 - 16x^2 + 4x.
    • Subtract this result from the current line.
         x^2 - 4x       
     ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2 
           -(x^4 + 4x^3 - x^2)
           --------------------
               -4x^3 - 16x^2 + 4x 
               -(-4x^3 - 16x^2 + 4x)
               --------------------
                       0 - 2

  6. Final steps:

    • Bring down the last term (-2).
    • We cannot divide -2 by x^2, so -2 becomes the remainder.
         x^2 - 4x       
     ___________
x^2+4x-1 | x^4 + 0x^3 - 17x^2 + 4x - 2 
           -(x^4 + 4x^3 - x^2)
           --------------------
               -4x^3 - 16x^2 + 4x 
               -(-4x^3 - 16x^2 + 4x)
               --------------------
                       0 - 2

Therefore, the result of the division is: (x^4 - 17x^2 + 4x - 2) / (x^2 + 4x - 1) = x^2 - 4x + (-2)/(x^2 + 4x - 1).

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