How do you find a third degree polynomial given roots #5# and #2i#?

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Answer 1 · 2016-11-07T21:11:32

Please see the explanation section below.

For a third degree polynomial, we need 3 linear factors.

Since #5# and #2i# are roots (zeros), we know that #x-5# and #x-2i# are factors.

If we want a polynomial with real coeficients, then the complex conjugate of #2i# (which is #-2i#) must also be a root and #x+2i# must be a factor.

One polynomial with real coefficients that meets the requirements is

#(x-5)(x-2i)(x+2i) = (x-5)(x^2+4)#

# = x^3-5x^2+4x-20#

Any constant multiple of this also meets the requirements.

For example

# 7(x^3-5x^2+4x-20) = 7x^3-35x^2+28x-140#