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2015-01-28T12:31:51+05:30

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To find the answer, you need to equate the value of the polynomial at polynomial to 0 at p=k.
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the given polynomial is f(p) = p³ - k²p + p + 2
if k is a root of the polynomial, 
then f(k) = 0
⇒ k³ - k²×k + k + 2 = 0
⇒ k³ - k³ + k + 2 = 0
⇒ k + 2 = 0
 k = -2

So the value of k is \boxed {-2}

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2015-01-28T16:36:40+05:30
F(p) = p³-k²p+p+2
       And K is a root of this polynomial,   p = k ⇒ p-k =0
⇒   p³-k²p+p+2 =0
⇒    p(p²-k²)+p+2 = 0
⇒     p(p-k)(p+k)+p+2  = 0
⇒ p.(0)(p+k)+p+2 = 0
⇒   p+2 = 0 
    ⇒     p = -2
and p = k,    
therefore K = -2

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