How do you write a polynomial function of minimum degree with real coefficients whose zeros include those listed: 5i and √5?

1 Answer
George C.
Nov 8, 2015

#f(x) = (x^2+25)(x-sqrt(5)) = x^3-sqrt(5)x^2+25-25sqrt(5)#

or if you want rational coefficients:

#g(x) = (x^2+25)(x^2-5) = x^4+20x^2-125#

Explanation:

If a polynomial has Real coefficients, then any non-Real Complex roots will occur in Complex conjugate pairs. So the roots of our polynomial must include #5i#, #-5i# and #sqrt(5)#.

If we allow irrational coefficients then the monic polynomial of lowest degree with these roots is:

#f(x) = (x-5i)(x+5i)(x-sqrt(5)) = (x^2+25)(x-sqrt(5))#

#= x^3-sqrt(5)x^2+25-25sqrt(5)#

To have rational coefficient then we also need the irrational conjugate #-sqrt(5)# of #sqrt(5)# resulting in:

#g(x) = (x-5i)(x+5i)(x-sqrt(5))(x+sqrt(5)) = (x^2+25)(x^2-5)#

#= x^4+20x^2-125#

Any polynomial with these roots will be a multiple (scalar or polynomial) of #f(x)# or #g(x)#

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