f(x)=(x-5)(x-2)(x-4)(x-4i)(x+4i)

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Solution for f(x)=(x-5)(x-2)(x-4)(x-4i)(x+4i) equation: 


Simplifying
f(x) = (x + -5)(x + -2)(x + -4)(x + -4i)(x + 4i)

Multiply f * x
fx = (x + -5)(x + -2)(x + -4)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-5 + x)(x + -2)(x + -4)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-5 + x)(-2 + x)(x + -4)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-5 + x)(-2 + x)(-4 + x)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-5 + x)(-2 + x)(-4 + x)(-4i + x)(x + 4i)

Reorder the terms:
fx = (-5 + x)(-2 + x)(-4 + x)(-4i + x)(4i + x)

Multiply (-5 + x) * (-2 + x)
fx = (-5(-2 + x) + x(-2 + x))(-4 + x)(-4i + x)(4i + x)
fx = ((-2 * -5 + x * -5) + x(-2 + x))(-4 + x)(-4i + x)(4i + x)
fx = ((10 + -5x) + x(-2 + x))(-4 + x)(-4i + x)(4i + x)
fx = (10 + -5x + (-2 * x + x * x))(-4 + x)(-4i + x)(4i + x)
fx = (10 + -5x + (-2x + x2))(-4 + x)(-4i + x)(4i + x)

Combine like terms: -5x + -2x = -7x
fx = (10 + -7x + x2)(-4 + x)(-4i + x)(4i + x)

Multiply (10 + -7x + x2) * (-4 + x)
fx = (10(-4 + x) + -7x * (-4 + x) + x2(-4 + x))(-4i + x)(4i + x)
fx = ((-4 * 10 + x * 10) + -7x * (-4 + x) + x2(-4 + x))(-4i + x)(4i + x)
fx = ((-40 + 10x) + -7x * (-4 + x) + x2(-4 + x))(-4i + x)(4i + x)
fx = (-40 + 10x + (-4 * -7x + x * -7x) + x2(-4 + x))(-4i + x)(4i + x)
fx = (-40 + 10x + (28x + -7x2) + x2(-4 + x))(-4i + x)(4i + x)
fx = (-40 + 10x + 28x + -7x2 + (-4 * x2 + x * x2))(-4i + x)(4i + x)
fx = (-40 + 10x + 28x + -7x2 + (-4x2 + x3))(-4i + x)(4i + x)

Combine like terms: 10x + 28x = 38x
fx = (-40 + 38x + -7x2 + -4x2 + x3)(-4i + x)(4i + x)

Combine like terms: -7x2 + -4x2 = -11x2
fx = (-40 + 38x + -11x2 + x3)(-4i + x)(4i + x)

Multiply (-40 + 38x + -11x2 + x3) * (-4i + x)
fx = (-40(-4i + x) + 38x * (-4i + x) + -11x2 * (-4i + x) + x3(-4i + x))(4i + x)
fx = ((-4i * -40 + x * -40) + 38x * (-4i + x) + -11x2 * (-4i + x) + x3(-4i + x))(4i + x)
fx = ((160i + -40x) + 38x * (-4i + x) + -11x2 * (-4i + x) + x3(-4i + x))(4i + x)
fx = (160i + -40x + (-4i * 38x + x * 38x) + -11x2 * (-4i + x) + x3(-4i + x))(4i + x)
fx = (160i + -40x + (-152ix + 38x2) + -11x2 * (-4i + x) + x3(-4i + x))(4i + x)
fx = (160i + -40x + -152ix + 38x2 + (-4i * -11x2 + x * -11x2) + x3(-4i + x))(4i + x)
fx = (160i + -40x + -152ix + 38x2 + (44ix2 + -11x3) + x3(-4i + x))(4i + x)
fx = (160i + -40x + -152ix + 38x2 + 44ix2 + -11x3 + (-4i * x3 + x * x3))(4i + x)
fx = (160i + -40x + -152ix + 38x2 + 44ix2 + -11x3 + (-4ix3 + x4))(4i + x)

Reorder the terms:
fx = (160i + -152ix + 44ix2 + -4ix3 + -40x + 38x2 + -11x3 + x4)(4i + x)
fx = (160i + -152ix + 44ix2 + -4ix3 + -40x + 38x2 + -11x3 + x4)(4i + x)

Multiply (160i + -152ix + 44ix2 + -4ix3 + -40x + 38x2 + -11x3 + x4) * (4i + x)
fx = (160i * (4i + x) + -152ix * (4i + x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = ((4i * 160i + x * 160i) + -152ix * (4i + x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))

Reorder the terms:
fx = ((160ix + 640i2) + -152ix * (4i + x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = ((160ix + 640i2) + -152ix * (4i + x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + (4i * -152ix + x * -152ix) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))

Reorder the terms:
fx = (160ix + 640i2 + (-152ix2 + -608i2x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + (-152ix2 + -608i2x) + 44ix2 * (4i + x) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + (4i * 44ix2 + x * 44ix2) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))

Reorder the terms:
fx = (160ix + 640i2 + -152ix2 + -608i2x + (44ix3 + 176i2x2) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + (44ix3 + 176i2x2) + -4ix3 * (4i + x) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + (4i * -4ix3 + x * -4ix3) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))

Reorder the terms:
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + (-4ix4 + -16i2x3) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + (-4ix4 + -16i2x3) + -40x * (4i + x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + (4i * -40x + x * -40x) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + (-160ix + -40x2) + 38x2 * (4i + x) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + (4i * 38x2 + x * 38x2) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + (152ix2 + 38x3) + -11x3 * (4i + x) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + 152ix2 + 38x3 + (4i * -11x3 + x * -11x3) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + 152ix2 + 38x3 + (-44ix3 + -11x4) + x4(4i + x))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + 152ix2 + 38x3 + -44ix3 + -11x4 + (4i * x4 + x * x4))
fx = (160ix + 640i2 + -152ix2 + -608i2x + 44ix3 + 176i2x2 + -4ix4 + -16i2x3 + -160ix + -40x2 + 152ix2 + 38x3 + -44ix3 + -11x4 + (4ix4 + x5))

Reorder the terms:
fx = (160ix + -160ix + -152ix2 + 152ix2 + 44ix3 + -44ix3 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)

Combine like terms: 160ix + -160ix = 0
fx = (0 + -152ix2 + 152ix2 + 44ix3 + -44ix3 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)
fx = (-152ix2 + 152ix2 + 44ix3 + -44ix3 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)

Combine like terms: -152ix2 + 152ix2 = 0
fx = (0 + 44ix3 + -44ix3 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)
fx = (44ix3 + -44ix3 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)

Combine like terms: 44ix3 + -44ix3 = 0
fx = (0 + -4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)
fx = (-4ix4 + 4ix4 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)

Combine like terms: -4ix4 + 4ix4 = 0
fx = (0 + 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)
fx = (640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5)

Solving
fx = 640i2 + -608i2x + 176i2x2 + -16i2x3 + -40x2 + 38x3 + -11x4 + x5

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by 'x'.
f = 640i2x-1 + -608i2 + 176i2x + -16i2x2 + -40x + 38x2 + -11x3 + x4

Simplifying
f = 640i2x-1 + -608i2 + 176i2x + -16i2x2 + -40x + 38x2 + -11x3 + x4

Reorder the terms:
f = -608i2 + 640i2x-1 + 176i2x + -16i2x2 + -40x + 38x2 + -11x3 + x4

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