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Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

 Jun 27, 2024
 #1
avatar+129725 
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a + b = 1    square both sides

 

a^2 + b^2 + 2ab = 1

 

     2  + 2ab = 1

 

2ab = -1

 

ab = -1/2

 

a^3 + b^3  =  (a + b) ( a^2 + b^2 -ab) = (1) (2 - -1/2) = 2 + 1/2  =  5/2

 

cool cool cool

 Jun 27, 2024

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