Another Dank Polynomial

Algebra Level 5

Let a , b , c , d a,b,c,d be real numbers such that b − d ≥ 5 b-d \ge 5 and all zeros x 1 , x 2 , x 3 , x_1, x_2, x_3, and x 4 x_4 of the polynomial P ( x ) = x 4 + a x 3 + b x 2 + c x + d P(x)=x^4+ax^3+bx^2+cx+d are real. Find the smallest value the product ( x 1 2 + 1 ) ( x 2 2 + 1 ) ( x 3 2 + 1 ) ( x 4 2 + 1 ) (x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) can take.


The answer is 16.

View wiki
Alan Yan by Alan Yan , 20, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Alan Yan
Aug 31, 2015

Consider the factorization: x k 2 + 1 = ( x k + i ) ( x k − i ) ⟹ ( x 1 2 + 1 ) ( x 2 2 + 1 ) ( x 3 2 + 1 ) ( x 4 2 + 1 ) = x_k^2+1 = (x_k+i)(x_k-i) \implies (x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) =

[ ( x 1 + i ) ( x 2 + i ) ( x 3 + i ) ( x 4 + i ) ] [ ( x 1 − i ) ( x 2 − i ) ( x 3 − i ) ( x 4 − i ) ] [(x_1+i)(x_2+i)(x_3+i)(x_4+i)][(x_1-i)(x_2-i)(x_3-i)(x_4-i)]

= [ ( − i − x 1 ) ( − i − x 2 ) ( − i − x 3 ) ( − i − x 4 ) ] [ ( i − x 1 ) ( i − x 2 ) ( i − x 3 ) ( i − x 4 ) ] = [(-i - x_1)(-i - x_2)(-i - x_3)(-i - x_4)][(i - x_1)(i-x_2)(i-x_3)(i-x_4)]

Since P ( x ) = ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) ( x − x 4 ) P(x) = (x - x_1)(x - x_2)(x-x_3)(x - x_4) , the expression is equivalent to

P ( i ) P ( − i ) = [ ( d − b + 1 ) + ( c − a ) i ] [ ( d − b + 1 ) − ( c − a ) i ] = P(i)P(-i) = [(d- b +1) +(c-a)i][(d-b+1)-(c-a)i] =

( b − d − 1 ) 2 + ( c − a ) 2 ≥ 16 (b-d-1)^2 + (c-a)^2 \geq \boxed{16}

Equality occurs when x i = 1 x_i = 1

This problem was from USAMO 2014

3 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...