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  4. Let α, β are roots of x2-3 x+5=0 then value of (5 α ⋅ β2016+3 α2017 ⋅ β+α ⋅ β2018/α2016+β2016)

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Q. Let $\alpha, \beta$ are roots of $x^2-3 x+5=0$ then value of $\frac{5 \alpha \cdot \beta^{2016}+3 \alpha^{2017} \cdot \beta+\alpha \cdot \beta^{2018}}{\alpha^{2016}+\beta^{2016}}$

Complex Numbers and Quadratic Equations

Solution:

$\alpha+\beta=3$ and $\alpha \beta=5$
$\frac{5 \alpha \beta^{2016}+3 \alpha^{2017} \beta+\alpha \beta^{2018}}{\alpha^{2016}+\beta^{2016}}=\frac{\alpha^2 \beta^{2017}+\alpha^{2018} \beta+\alpha^{2017} \beta^2+\alpha \beta^{2018}}{\alpha^{2016}+\beta^{2016}}$
$=\frac{\alpha^2 \beta\left(\alpha^{2016}+\beta^{2016}\right)+\beta^2 \alpha\left(\alpha^{2016}+\beta^{2016}\right)}{\alpha^{2016}+\beta^{2016}}=\alpha^2 \beta+\beta^2 \alpha=15$