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

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

Simplifying the Expression: (x^5 - 4x^3 + 4x^2) / (x - 4)

This expression represents a rational function, where a polynomial (x^5 - 4x^3 + 4x^2) is divided by a linear expression (x - 4). To simplify this expression, we can utilize polynomial long division.

Here's how the process works:

  1. Set up the division:

         ___________
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0

    We add placeholder terms (0x and 0) for missing powers of x in the dividend.

  2. Divide the leading terms:

    • The leading term of the dividend (x^5) is divided by the leading term of the divisor (x), resulting in x^4.
    • Write x^4 above the division line.
         x^4______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0

  3. Multiply the divisor by the quotient:

    • Multiply (x - 4) by x^4, which gives x^5 - 4x^4.
    • Write this result below the dividend, aligning terms by their powers.
         x^4______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4

  4. Subtract:

    • Subtract the result from the dividend, changing signs of the terms in the bottom line.
         x^4______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3

  5. Bring down the next term:

    • Bring down the next term from the dividend (+4x^2).
         x^4______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3 + 4x^2

  6. Repeat steps 2-5:

    • Divide the new leading term (4x^4) by the divisor's leading term (x), resulting in 4x^3.
    • Multiply (x - 4) by 4x^3 and write the result below.
    • Subtract the result.
    • Bring down the next term (0x).
         x^4 + 4x^3______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3 + 4x^2
           4x^4 - 16x^3 
           --------
                    12x^3 + 4x^2

  7. Continue the process:

    • Repeat steps 2-5 until the degree of the remainder is less than the degree of the divisor.
         x^4 + 4x^3 + 12x^2______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3 + 4x^2
           4x^4 - 16x^3 
           --------
                    12x^3 + 4x^2 + 0x
                    12x^3 - 48x^2
                    --------
                             52x^2 + 0x

         x^4 + 4x^3 + 12x^2 + 52x______
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3 + 4x^2
           4x^4 - 16x^3 
           --------
                    12x^3 + 4x^2 + 0x
                    12x^3 - 48x^2
                    --------
                             52x^2 + 0x + 0
                             52x^2 - 208x
                             --------
                                     208x + 0

         x^4 + 4x^3 + 12x^2 + 52x + 208
x-4 | x^5 - 4x^3 + 4x^2 + 0x + 0 
      x^5 - 4x^4 
      --------
           4x^4 - 4x^3 + 4x^2
           4x^4 - 16x^3 
           --------
                    12x^3 + 4x^2 + 0x
                    12x^3 - 48x^2
                    --------
                             52x^2 + 0x + 0
                             52x^2 - 208x
                             --------
                                     208x + 0
                                     208x - 832
                                     --------
                                              832

  8. Result:

    • The quotient is x^4 + 4x^3 + 12x^2 + 52x + 208.
    • The remainder is 832.

Therefore, the simplified form of the expression (x^5 - 4x^3 + 4x^2) / (x - 4) is:

x^4 + 4x^3 + 12x^2 + 52x + 208 + 832 / (x - 4)

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