Answer: f(x) = 3x^3 - 15x^2 + 12x - 60
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Explanation:
Let's focus on the root x = 2i. Square both sides to get rid of the imaginary component and then get everything to one side.
x = 2i
x^2 = (2i)^2
x^2 = 4i^2
x^2 = -4
x^2+4 = 0
You should find that the root x = -2i leads to x^2+4 = 0 as well.
So the equation x^2+4 = 0 produces the complex roots x = 2i and x = -2i. This tells us that x^2+4 is a factor of the answer. The other factor is x-5 due to the real number root x = 5.
We'll multiply those two factors mentioned:
(x-5)(x^2+4)
w(x^2+4) ..... let w = x-5
wx^2 + 4w
x^2(w) + 4(w)
x^2(x-5) + 4(x-5) .... plug in w = x-5
x^3 - 5x^2 + 4x - 20
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The last step is to triple each term so that we go from the leading term of x^3 to 3x^3. This will give us a lead coefficient of 3.
So we go from this
x^3 - 5x^2 + 4x - 20
to this
3x^3 - 15x^2 + 12x - 60
after tripling everything. This expression has the same roots as the previous one. Scaling the function only stretches it vertically, but the roots aren't changed.
