How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are i, -3i, 3i?

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Answer 1 · 2016-07-11T00:15:12

#x^3-x^2i+9x-9i=0#

When we talk about the zeros of a polynomial, we can express them as #x="the zero"#. So with the terms listed above, we can say:

#x=i, -3i, 3i# and we can therefore say:

#x-i=0, x+3i=0, x-3i=0#

which leads to:

#(x-i)(x+3i)(x-3i)=0#

Expanding and collecting like terms for coefficients in the cubic

#x^3-x^2i+9x-9i=0#.

Also, we can use the form

#x^3#-

(sum of the roots)x^2+

sum of the products of the roots, taken two at a time)x-

product of the roots

= 0.

Here, it is

#x^3-(i+3i-3i)x^2+(3-3+9)x-(9i)=0#. Simplifying,

#x^3-x^2i+9x-9i=0#