Question

If a, b, c are three real roots of the equation x3−6x2+5x−1=0x3-6x2+5x-1=0, then the equation, whose roots a2,b2,c2a2,b2,c2 will be ​

Answer 1

Given : a, b, c are three real roots of the equation x³  - 6x² + 5x − 1 = 0

To find : the equation, whose roots are a² , b² , c²

Solution:

a, b, c are three real roots of the equation  

x³  - 6x² + 5x − 1 = 0

a + b  + c  =  -(-6)/1 = 6

ab + bc + ca  = 5/1  = 5

abc   = -(-1)/1 =  1

a² , b² , c²   are  roots so need to find

a² + b² + c²     , a²b² + a²c² + b²c²    , a²b²c²

as x³ - (a² + b² + c²)x² + x( a²b² + a²c² + b²c²) -  a²b²c²  = 0  is the equation

a²b²c²  = (abc)²   = 1

(a + b  + c)²  = a² + b² + c² + 2(ab + bc + ca )

=> 6² = a² + b² + c² + 2(5)

=> a² + b² + c² = 26

ab + bc + ca  = 5

Squaring both sides

=> a²b² + a²c² + b²c²  + 2(ab²c + a²bc + abc²)  = 25

=>  a²b² + a²c² + b²c² + 2abc(b + a + c) = 25

=> a²b² + a²c² + b²c² + 2 (6) = 25

=> a²b² + a²c² + b²c² = 13

a² + b² + c² = 26     , a²b² + a²c² + b²c² = 13   , a²b²c²  = 1

x³  - 26x² + 13x − 1 = 0

x³  - 26x² + 13x − 1 = 0 is the equation  having roots a² , b² , c²

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