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Algebra Level 4

Let a , b , c a,b,c be distinct non zero real numbers such that a + 1 / b = b + 1 / c = c + 1 / a a+1/b=b+1/c=c+1/a Then which of the following polynomial has a root which is equal to|abc|

3 x 2 − 7 x + 2 3x^2-7x+2 x 2 − 110 x + 1000 x^2-110x+1000 x 2 − 10 x + 9 x^2-10x+9 x 3 − 12 x 2 + 47 x − 60 x^3-12x^2+47x-60 72 x 2 − 78 x + 1 72x^2-78x+1

Ravi Dwivedi by Ravi Dwivedi , 24, India

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1 solution

Ravi Dwivedi
Jul 9, 2015

a + 1 / b = b + 1 / c ⟹ a − b = b − c b c a+1/b=b+1/c \implies a-b=\frac {b-c}{bc}

Similarly from other two equations b − c = c − a c a a n d c − a = a − b a b b-c=\frac {c-a}{ca} and c-a=\frac {a-b}{ab}

Multiplying these three equations we get ( a − b ) ( b − c ) ( c − a ) = ( a − b ) ( b − c ) ( c − a ) ( a b c ) 2 (a-b)(b-c)(c-a)=\frac{(a-b)(b-c)(c-a)}{(abc)^{2}}

Cancelling ( a − b ) ( b − c ) ( c − a ) (a-b)(b-c)(c-a) as a,b,c are distinct and nonzero so this product becomes nonzero in its domain. We get ( a b c ) 2 = 1 (abc)^{2}=1 ⟹ ∣ a b c ∣ = 1 \implies |abc|=1

and 1 1 is a root of the equation x 2 − 10 x + 9 x^{2}-10x+9 only among the given options.

2 Helpful 1 Interesting 1 Brilliant 0 Confused

Moderator note:

Does there exist 3 numbers that satisfy the conditions in the question? If so, how can we classify them?

FYI You can Latex the answer choices too! I've edited these for you :)

Calvin Lin Staff - 5 years, 11 months ago

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Thank you!!

Ravi Dwivedi - 5 years, 11 months ago

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