Multiple choice

If there is a root of order $3$ for the equation $x^{4}-2x^{3}+2x-1=0$, then the other root is:

  1. $-1$
  2. $0$
  3. $1$
  4. $2$
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A Correct answer
Explanation

The equation is x^4 - 2x^3 + 2x - 1 = 0. Factoring, we get (x-1)^3 * (x+1) = 0. The root of order 3 is x=1, so the other root is -1.

AI explanation

If a polynomial has a root of order 3, it can be written as (x - c)^3 multiplied by a linear factor. Applying binomial expansion to (x - 1)^3 gives x^3 - 3x^2 + 3x - 1, which is a factor of the given polynomial x^4 - 2x^3 + 2x - 1. Performing polynomial division yields the other linear factor as x + 1. Setting x + 1 = 0 gives the other root as -1.

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