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Simplifying z^{7}+ 2(z^{4}) + 4z = 0 Reorder the terms: 4z + 2z^{4}+ z^{7}= 0 Solving 4z + 2z^{4}+ z^{7}= 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(4 + 2z^{3}+ z^{6}) = 0## Subproblem 1

Set the factor 'z' equal to zero and attempt to solve: Simplifying z = 0 Solving z = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z = 0## Subproblem 2

Set the factor '(4 + 2z^{3}+ z^{6})' equal to zero and attempt to solve: Simplifying 4 + 2z^{3}+ z^{6}= 0 Solving 4 + 2z^{3}+ z^{6}= 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + 2z^{3}+ -4 + z^{6}= 0 + -4 Reorder the terms: 4 + -4 + 2z^{3}+ z^{6}= 0 + -4 Combine like terms: 4 + -4 = 0 0 + 2z^{3}+ z^{6}= 0 + -4 2z^{3}+ z^{6}= 0 + -4 Combine like terms: 0 + -4 = -4 2z^{3}+ z^{6}= -4 The z term is 2z^{3}. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2z^{3}+ 1 + z^{6}= -4 + 1 Reorder the terms: 1 + 2z^{3}+ z^{6}= -4 + 1 Combine like terms: -4 + 1 = -3 1 + 2z^{3}+ z^{6}= -3 Factor a perfect square on the left side: (z^{3}+ 1)(z^{3}+ 1) = -3 Can't calculate square root of the right side. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.## Solution

z = {0}

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