If α , β are the roots of x 2 − 6 x + 5 = 0 , then the value of α 3 β + α β 3 is:

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Given the equation x 2 − 6 x + 5 = 0 , with roots α , β . By Vieta's formulas, α + β = 6 and α β = 5 . We need to find the value of α 3 β + α β 3 . This expression can be factored as α β ( α 2 + β 2 ) . We know α 2 + β 2 = ( α + β ) 2 − 2 α β . Substituting the values: α 2 + β 2 = ( 6 ) 2 − 2 ( 5 ) = 36 − 10 = 26 . Therefore, α β ( α 2 + β 2 ) = 5 ( 26 ) = 130 .

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