Problem 4

A polynomial f ( *) has the factor - square property ( or ESP ) if f ( * ) is a factor of f ( 20 2 ) .

For instance ,* 9 ( 20 ) = * - 1 and h ( a ) = ac have ESP , but K ( 20 ) = * + 2 does not .

Reason : * - 1 is a factor of 2 2 - 1 , and * is a factor of *2 , but a + 2 is not a factor of 2 2 + 2 .

Multiplying by a nonzero constant " preserves " ISP , so we restrict attention to poly -

nomials that are monic ( i.e ., have I as highest- degree coefficient ) .

What is the pattern to these ESP polynomials ? To make progress on this general

problem , investigate the following questions and justify your answers .

( a ) Are * and * - I the only monic polynomials of degree I with ESP ?

( b ) Check that 2 2 , 2 2 - 1 , 2 2 - * , and a 2 + 2 + 1 all have ESP. Determine all the

monic degree 2 polynomials with ESP .

( c ) Some of our examples are products of ESP polynomials of smaller degree . For

instance , * _ and a_ _ * come from degree I cases . However , 2 2 - 1 and 2 2 + 2 + 1

are new , not expressible as a product of two smaller ISP polynomials .

Are there monic ESP polynomials of degree & that are new ( not built from ESP

polynomials of smaller degree ) ?'

Are there such examples of degree 4 ?

( d ) The examples written above all had integer coefficients . Do answers change if we

allow polynomials whose coefficients are allowed to be any real numbers ? Or if

we allow polynomials whose coefficients are complex numbers ?'