If alpha and beeta are the …
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Sia ? 6 years, 6 months ago
Given polynomial is
f(x) = x2 + px + q
Sum of the zeroes = {tex}α + β = -p{/tex}
Product of the zeroes = {tex}αβ = q{/tex}
As per given condition
Sum of the zeroes of new polynomial = {tex}(α+β)^2 + (α-β)^2{/tex}
= {tex}(α+β)^2 + α^2 + β^2 - 2αβ{/tex}
= {tex}(α+β)^2 + (α+β)^2 - 2αβ - 2αβ{/tex}
= {tex}(-p)^2 + (-p)^2 - 2 × q - 2 × q{/tex}
= {tex}p^2 + p^2 - 4q{/tex}
= {tex}2p^2 - 4q{/tex}
Product of the zeroes of new polynomial ={tex} (α + β)^2 (α - β)^2{/tex}
= {tex}(-p)^2((-p)^2 - 4q){/tex}
= {tex}p^2(p^2 - 4q){/tex}
So, the quadratic polynomial is,
x2 - (sum of the zeroes)x + (product of the zeroes)
= x2 - (2p2 - 4q)x + p2(p2 - 4q)
Hence, the required quadratic polynomial is f(x) = x2 - (2p2 - 4q)x + p2(p2 - 4q).
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