Since \( p(1) = 0 \), \( x = 1 \) is indeed a root. To check the multiplicity, we compute the derivative \( p'(x) \):
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- Solution:** To verify if \( x = 1 \) is a root of multiplicity greater than 1 for \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), we first check if \( p(1) = 0 \).
- Calculate \( p(1) \):
- p(1) = 1^4 - 4 imes 1^3 + 6 imes 1^2 - 4 imes 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0
- p'(x) = 4x^3 - 12x^2 + 12x - 4
- Now, evaluate \( p'(1) \):
- p'(1) = 4 imes 1^3 - 12 imes 1^2 + 12 imes 1 - 4 = 4 - 12 + 12 - 4 = 0
