Consider the polynomial \(p(x) = x^5+2x^4+x^3-x^2-2x-1\) : I. \(x^2+2x+1\) is a factor of p(x). II. \(x^2+x+1\) divides p(x). Which of the statements given above is/are correct?
Both I and II
Dividing \(p(x)\) by \(x^2+2x+1=(x+1)^2\) gives quotient \(x^3-1\) with zero remainder, so \(p(x) = (x+1)^2(x^3-1) = (x+1)^2(x-1)(x^2+x+1)\). This shows \(x^2+2x+1\) is indeed a factor (statement I true), and \(x^2+x+1\) also divides \(p(x)\) exactly (statement II true). Hence both I and II are correct.
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