rzfansubs87

rzfansubs87

Answered

2022-07-01

If a b and y are the roots of 3 x 3 + 8 x 2 1 = 0 find ( b + 1 / y ) ( y + 1 / a ) ( a + 1 / b )

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Answer & Explanation

Sophia Mcdowell

Sophia Mcdowell

Expert

2022-07-02Added 14 answers

You have a few mistakes:
( b + 1 y ) ( y + 1 a ) ( a + 1 b ) a 2 b 2 y 2 + 1 a b y , as you write above
( b + 1 y ) ( y + 1 a ) ( a + 1 b ) a 2 b 2 y 2 + 1 a b y , as you apply the formula you wrote
The actual solution is:
( b + 1 y ) ( y + 1 a ) ( a + 1 b ) = 1 a b y ( y b + 1 ) ( a y + 1 ) ( a b + 1 ) = 1 a b y ( a b y 2 + a y + y b + 1 ) ( a b + 1 ) = 1 a b y ( a 2 b 2 y 2 + a 2 b y + a b 2 y + a b y 2 + a b + a y + y b + 1 ) = a b y + a + b + y + a b + a y + y b a b y + 1 a b y
Now, using Vieta's Formulas, we have that a b y = 1 3 , a b + a y + y b = 0 , a + b + y = 8 3 . This gives ( b + 1 y ) ( y + 1 a ) ( a + 1 b ) = 1 3 8 3 + 0 + 3 = 2 3

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Desirae Washington

Desirae Washington

Expert

2022-07-03Added 5 answers

You can also calculate the expression directly using the fact that
p ( x ) = 3 ( x a ) ( x b ) ( x y ) 3 a b y = 1
To do so, note that P = ( b + 1 / y ) ( y + 1 / a ) ( a + 1 / b ) = ( b y + 1 ) ( a y + 1 ) ( a b + 1 ) a b y = a b y = 1 3 3 ( 1 3 a + 1 ) ( 1 3 b + 1 ) ( 1 3 y + 1 ) = a b y = 1 3 1 3 ( 1 + 3 a ) ( 1 + 3 b ) ( 1 + 3 y )
Now, a standard trick uses the observation
p ( t 1 3 ) = 3 ( t 1 3 a ) ( t 1 3 b ) ( t 1 3 y ) = 1 9 ( t ( 1 + 3 a ) ) ( t ( 1 + 3 b ) ) ( t ( 1 + 3 y ) )
So, you only need the constant member
c = 1 9 ( 1 + 3 a ) ( 1 + 3 b ) ( 1 + 3 y ) = 1 3 P
of
p ( t 1 3 ) = 3 ( t 1 3 ) 3 + 8 ( t 1 3 ) 2 1
Hence, P = 3 c = 2 3

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