Question

if alpha and beta are the zeros of a quadratic polynomial f(x)=x^2-4x+3, find the value of(alpha^4×beta^2+alpha^2×beta^4)

Answer 1

$if \: \alpha \: and \: \beta \: are \: zeros \: \\ then from the question \: \alpha + \beta = 4 \: \: and \: \: \: \alpha \beta = 3 \\ \\ now \: \: \: { \alpha }^{4} { \beta }^{2} + { \alpha }^{2} { \beta }^{4} = { \alpha }^{2} { \beta }^{2} ( { \alpha }^{2} + { \beta }^{2} ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ = \ { (\alpha \beta )}^{2} {(( { \alpha + \beta) }^{2} - 2 \alpha \beta }) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 9 \times (16 - 6) = 90.$



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