Polynomial propaganda!

Algebra Level 3

Let a , b , c , d , e , f a,b,c,d,e,f be constant real numbers such that x 8 − 4 x 7 + 7 x 6 + a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0 x^8-4x^7+7x^6 + ax^5 + bx^4 + cx^3 + dx^2+ex + f = 0 has 8 real roots.

Find f . f.

256 256 1 16 \frac1{16} 16 16 1 256 \frac1{256}

Rudrayan Kundu by Rudrayan Kundu , 16, India

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1 solution

Rudrayan Kundu
Nov 6, 2018

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Slight technicality.

Note that what you gave is a necessary, but not sufficient (as yet), proof. You still need to show that for 8 linear factors with x i = x j = 1 2 x_i = x_j = \frac{1}{2} , the conditions on the coefficients are satisfied.

(Yes, your proof contains enough information to fill in the gap, which is why it's more of a technicality than a gap.)

Calvin Lin Staff - 2 years, 7 months ago

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