Polynomials

Algebra Level 5

If the roots of the polynomial 2 x 3 − 4 x 2 + 7 x − 5 = 0 2x^{3}-4x^{2}+7x-5=0 are a , b , c a,b,c then find the value of ∑ c y c a , b , c 1 ( b − 4 ) ( a − 2 ) + 2 a − 4 \sum_{cyc}^{a,b,c}\frac{1}{(b-4)(a-2)+2a-4}

Source: Tallentex.


The answer is 0.888.

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Shivam Jadhav by Shivam Jadhav , 22, India

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3 solutions

Siddharth Singh
Oct 14, 2015

Simplifying the expression we get:

1 ( a − 2 ) ( b − 2 ) + 1 ( b − 2 ) ( c − 2 ) + 1 ( c − 2 ) ( a − 2 ) = a + b + c − 6 a b c − 2 ( a b + b c + c a ) + 4 ( a + b + c ) − 8 \frac{1}{(a-2)(b-2)}+\frac{1}{(b-2)(c-2)}+\frac{1}{(c-2)(a-2)}=\frac{a+b+c-6}{abc-2(ab+bc+ca)+4(a+b+c)-8}

from the polynomial: a + b + c = 2 a+b+c=2

a b + b c + c a = 7 2 ab+bc+ca=\frac{7}{2}

a b c = 5 2 abc=\frac{5}{2}

Substituting the values in the expression we get the answer as 0.8888 \boxed{0.8888}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly, what I did.

Dev Sharma - 5 years, 8 months ago

If you write (a-2) as (2-a) and like that for all the roots in the expression...and make the necessary sign changes then you can put x=2 in the polynomial which will be your denominator. The calculation will be simpler.

Rudrayan Kundu - 2 years, 11 months ago
Aakash Khandelwal
Oct 15, 2015

Simplification tool: one root is unity ie. 1

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Lu Chee Ket
Oct 15, 2015

Using Complex functions, just sum up for (a, b, c), (c, a, b) and (b, c, a):

0.333333333333333+0.333333333333333i

0.333333333333333-0.333333333333333i

0.222222222222222

Answer: 0.888888888888888

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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