Vieta's Formula 2
Let α and β denote the roots of 3 x 2 + 3 x − 5 = 0 . What is the value of α 3 + β 3 ?
The answer is -6.
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2 solutions
3 x 2 + 3 x − 5 = 0 ⇒ 3 x 3 = − 3 x 2 + 5 x = ( 3 x − 5 ) + 5 x = 8 x − 5
Thus, { 3 α 3 = 8 α − 5 3 β 3 = 8 β − 5 ⇒ 3 ( α 3 + β 3 ) = 8 ( α + β ) − 1 0 ⇒ α 3 + β 3 = 3 1 ( 8 ⋅ − 3 3 − 1 0 ) = − 6 .
α + β = − 1 α β = − 3 5 α 3 + β 3 = ( α + β ) 3 − 3 α β ( α + β ) = ( − 1 ) 3 − 3 × − 3 5 ( − 1 ) = − 6