How do you write a polynomial function given the real zeroes -2,-2,3,-4i and coefficient 1?

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Answer 1 · 2016-01-05T03:07:17

#x^5+x^4+8 x^3+4 x^2-128 x-192#

The major trick with this problem is remembering that complex roots always come in pairs.

Thus, along with the root of #-4i#, the polynomial will also have a root of #4i#.

The polynomial can be written as:

#(x+2)^2(x-3)(x+4i)(x-4i)#

#=(x^2+4x+4)(x-3)(x^2+16)#

When distributed completely, this gives

#x^5+x^4+8 x^3+4 x^2-128 x-192#

graph{x^5+x^4+8 x^3+4 x^2-128 x-192 [-10, 10, -500, 301.6]}

As you can see, the graph has an odd degree #(5)#, has a root of #-2# with multiplicity #2# and a root at #3#.