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Simplifying 0 = (4 + -1z)(z + 5)(z + 4)(z + -5) Reorder the terms: 0 = (4 + -1z)(5 + z)(z + 4)(z + -5) Reorder the terms: 0 = (4 + -1z)(5 + z)(4 + z)(z + -5) Reorder the terms: 0 = (4 + -1z)(5 + z)(4 + z)(-5 + z) Multiply (4 + -1z) * (5 + z) 0 = (4(5 + z) + -1z * (5 + z))(4 + z)(-5 + z) 0 = ((5 * 4 + z * 4) + -1z * (5 + z))(4 + z)(-5 + z) 0 = ((20 + 4z) + -1z * (5 + z))(4 + z)(-5 + z) 0 = (20 + 4z + (5 * -1z + z * -1z))(4 + z)(-5 + z) 0 = (20 + 4z + (-5z + -1z2))(4 + z)(-5 + z) Combine like terms: 4z + -5z = -1z 0 = (20 + -1z + -1z2)(4 + z)(-5 + z) Multiply (20 + -1z + -1z2) * (4 + z) 0 = (20(4 + z) + -1z * (4 + z) + -1z2 * (4 + z))(-5 + z) 0 = ((4 * 20 + z * 20) + -1z * (4 + z) + -1z2 * (4 + z))(-5 + z) 0 = ((80 + 20z) + -1z * (4 + z) + -1z2 * (4 + z))(-5 + z) 0 = (80 + 20z + (4 * -1z + z * -1z) + -1z2 * (4 + z))(-5 + z) 0 = (80 + 20z + (-4z + -1z2) + -1z2 * (4 + z))(-5 + z) 0 = (80 + 20z + -4z + -1z2 + (4 * -1z2 + z * -1z2))(-5 + z) 0 = (80 + 20z + -4z + -1z2 + (-4z2 + -1z3))(-5 + z) Combine like terms: 20z + -4z = 16z 0 = (80 + 16z + -1z2 + -4z2 + -1z3)(-5 + z) Combine like terms: -1z2 + -4z2 = -5z2 0 = (80 + 16z + -5z2 + -1z3)(-5 + z) Multiply (80 + 16z + -5z2 + -1z3) * (-5 + z) 0 = (80(-5 + z) + 16z * (-5 + z) + -5z2 * (-5 + z) + -1z3 * (-5 + z)) 0 = ((-5 * 80 + z * 80) + 16z * (-5 + z) + -5z2 * (-5 + z) + -1z3 * (-5 + z)) 0 = ((-400 + 80z) + 16z * (-5 + z) + -5z2 * (-5 + z) + -1z3 * (-5 + z)) 0 = (-400 + 80z + (-5 * 16z + z * 16z) + -5z2 * (-5 + z) + -1z3 * (-5 + z)) 0 = (-400 + 80z + (-80z + 16z2) + -5z2 * (-5 + z) + -1z3 * (-5 + z)) 0 = (-400 + 80z + -80z + 16z2 + (-5 * -5z2 + z * -5z2) + -1z3 * (-5 + z)) 0 = (-400 + 80z + -80z + 16z2 + (25z2 + -5z3) + -1z3 * (-5 + z)) 0 = (-400 + 80z + -80z + 16z2 + 25z2 + -5z3 + (-5 * -1z3 + z * -1z3)) 0 = (-400 + 80z + -80z + 16z2 + 25z2 + -5z3 + (5z3 + -1z4)) Combine like terms: 80z + -80z = 0 0 = (-400 + 0 + 16z2 + 25z2 + -5z3 + 5z3 + -1z4) 0 = (-400 + 16z2 + 25z2 + -5z3 + 5z3 + -1z4) Combine like terms: 16z2 + 25z2 = 41z2 0 = (-400 + 41z2 + -5z3 + 5z3 + -1z4) Combine like terms: -5z3 + 5z3 = 0 0 = (-400 + 41z2 + 0 + -1z4) 0 = (-400 + 41z2 + -1z4) Solving 0 = -400 + 41z2 + -1z4 Solving for variable 'z'. Combine like terms: 0 + 400 = 400 400 + -41z2 + z4 = -400 + 41z2 + -1z4 + 400 + -41z2 + z4 Reorder the terms: 400 + -41z2 + z4 = -400 + 400 + 41z2 + -41z2 + -1z4 + z4 Combine like terms: -400 + 400 = 0 400 + -41z2 + z4 = 0 + 41z2 + -41z2 + -1z4 + z4 400 + -41z2 + z4 = 41z2 + -41z2 + -1z4 + z4 Combine like terms: 41z2 + -41z2 = 0 400 + -41z2 + z4 = 0 + -1z4 + z4 400 + -41z2 + z4 = -1z4 + z4 Combine like terms: -1z4 + z4 = 0 400 + -41z2 + z4 = 0 Factor a trinomial. (16 + -1z2)(25 + -1z2) = 0 Factor a difference between two squares. ((4 + z)(4 + -1z))(25 + -1z2) = 0 Factor a difference between two squares. ((5 + z)(5 + -1z))(4 + z)(4 + -1z) = 0Subproblem 1
Set the factor '(4 + z)' equal to zero and attempt to solve: Simplifying 4 + z = 0 Solving 4 + z = 0 Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + z = 0 + -4 Combine like terms: 4 + -4 = 0 0 + z = 0 + -4 z = 0 + -4 Combine like terms: 0 + -4 = -4 z = -4 Simplifying z = -4Subproblem 2
Set the factor '(4 + -1z)' equal to zero and attempt to solve: Simplifying 4 + -1z = 0 Solving 4 + -1z = 0 Move all terms containing z to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + -1z = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -1z = 0 + -4 -1z = 0 + -4 Combine like terms: 0 + -4 = -4 -1z = -4 Divide each side by '-1'. z = 4 Simplifying z = 4Subproblem 3
Set the factor '(5 + z)' equal to zero and attempt to solve: Simplifying 5 + z = 0 Solving 5 + z = 0 Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + z = 0 + -5 Combine like terms: 5 + -5 = 0 0 + z = 0 + -5 z = 0 + -5 Combine like terms: 0 + -5 = -5 z = -5 Simplifying z = -5Subproblem 4
Set the factor '(5 + -1z)' equal to zero and attempt to solve: Simplifying 5 + -1z = 0 Solving 5 + -1z = 0 Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + -1z = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -1z = 0 + -5 -1z = 0 + -5 Combine like terms: 0 + -5 = -5 -1z = -5 Divide each side by '-1'. z = 5 Simplifying z = 5Solution
z = {-4, 4, -5, 5}
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