How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 1, 1, i, -i?

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Answer 1 · 2016-11-20T17:19:59

Function is #x^4-2x^3+2x^2-2x+1#

A function with leading coefficient as #1# and zeros as #a#, #b#, #c# and #d# is

#f(x)=(x-a)(x-b)(x-c)(x-d)#

Hence if zeros are #1#, #1#, #i# and #-i#, the function is

#f(x)=(x-1)(x-1)(x-i)(x-(-i))#

= #(x^2-2x+1)(x-i)(x+i)#

= #(x^2-2x+1)(x^2-i^2)#

= #(x^2-2x+1)(x^2+1)# (as #i^2=-1#)

= #x^4-2x^3+x^2+x^2-2x+1#

= #x^4-2x^3+2x^2-2x+1#