Answer :

We know that,


p(x) = g(x)q(x) + r(x)


Putting values, we get


x3 – 3x2 + x + 2 = g(x)(x – 2) – 2x + 4


g(x)(x – 2) = x3 – 3x2 + x + 2 + 2x – 4


g(x)(x – 2) = x3 – 3x2 + 3x – 2


g(x)(x – 2) = x3 – 2x2 – x2 + 2x + x – 2


g(x)(x – 2) = x2(x – 2) – x(x – 2) + x – 2


g(x)(x – 2) = (x – 2)(x2 – x + 1)


g(x) = x2 – x + 1


OR


Let zeroes of x2 – 8x + k = 40 be α and β


α + β = 8 [-b/a]


αβ = k [c/a]


Given,


α2 + β2 = 40


(α + β)2 – 2αβ = 40


82 – 2k = 40


64 – 2k = 40


2k = 24


k = 12


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