Question

please answer this question (2)

with steps

Answer 1

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Answer 2

$given \: \: one \: \: zero = 1 + \sqrt{5} \\ then \: \: other \: \: zero = 1 - \sqrt{5} \\ \\ (reason \: \: is \: \: irrational \: \: zeros \\ of \: \: quadratic \: \: polynomial \: \: is \\ conjugate \: \: to \: \: each \: \: other) \\ \\ sum \: \: of \: \: zeros \\ = (1 + \sqrt{5} ) + (1 - \sqrt{5} ) = 2 \\ product \: \: of \: \: zeros \\ = (1 + \sqrt{5} ) \times (1 - \sqrt{5} ) = 1 - 5 = - 4 \\ \\ therefore \: \: polynomial \: \: is \\p(x) = k( {x}^{2} + (sum \: of \: zeros)x + (product \: of \: zeros) )\\ = k({x}^{2} + 2x - 4) \\ \\ given \: \: p(1) = 2 \\ k( {1}^{2} + 2 \times 1 - 4) = 2 \\ k \times - 1 = 2 \\ k = - 2 \\ \\ therefore \: \: required \: \: polynomial \: \: is \\ p(x) = - 2 {x}^{2} - 4x + 8$