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Simplifying f(x) = (x + -2)(x + -5)(x + 5i)(x + -5i) Multiply f * x fx = (x + -2)(x + -5)(x + 5i)(x + -5i) Reorder the terms: fx = (-2 + x)(x + -5)(x + 5i)(x + -5i) Reorder the terms: fx = (-2 + x)(-5 + x)(x + 5i)(x + -5i) Reorder the terms: fx = (-2 + x)(-5 + x)(5i + x)(x + -5i) Reorder the terms: fx = (-2 + x)(-5 + x)(5i + x)(-5i + x) Multiply (-2 + x) * (-5 + x) fx = (-2(-5 + x) + x(-5 + x))(5i + x)(-5i + x) fx = ((-5 * -2 + x * -2) + x(-5 + x))(5i + x)(-5i + x) fx = ((10 + -2x) + x(-5 + x))(5i + x)(-5i + x) fx = (10 + -2x + (-5 * x + x * x))(5i + x)(-5i + x) fx = (10 + -2x + (-5x + x2))(5i + x)(-5i + x) Combine like terms: -2x + -5x = -7x fx = (10 + -7x + x2)(5i + x)(-5i + x) Multiply (10 + -7x + x2) * (5i + x) fx = (10(5i + x) + -7x * (5i + x) + x2(5i + x))(-5i + x) fx = ((5i * 10 + x * 10) + -7x * (5i + x) + x2(5i + x))(-5i + x) fx = ((50i + 10x) + -7x * (5i + x) + x2(5i + x))(-5i + x) fx = (50i + 10x + (5i * -7x + x * -7x) + x2(5i + x))(-5i + x) fx = (50i + 10x + (-35ix + -7x2) + x2(5i + x))(-5i + x) fx = (50i + 10x + -35ix + -7x2 + (5i * x2 + x * x2))(-5i + x) fx = (50i + 10x + -35ix + -7x2 + (5ix2 + x3))(-5i + x) Reorder the terms: fx = (50i + -35ix + 5ix2 + 10x + -7x2 + x3)(-5i + x) fx = (50i + -35ix + 5ix2 + 10x + -7x2 + x3)(-5i + x) Multiply (50i + -35ix + 5ix2 + 10x + -7x2 + x3) * (-5i + x) fx = (50i * (-5i + x) + -35ix * (-5i + x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = ((-5i * 50i + x * 50i) + -35ix * (-5i + x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) Reorder the terms: fx = ((50ix + -250i2) + -35ix * (-5i + x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = ((50ix + -250i2) + -35ix * (-5i + x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + (-5i * -35ix + x * -35ix) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) Reorder the terms: fx = (50ix + -250i2 + (-35ix2 + 175i2x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + (-35ix2 + 175i2x) + 5ix2 * (-5i + x) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + (-5i * 5ix2 + x * 5ix2) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) Reorder the terms: fx = (50ix + -250i2 + -35ix2 + 175i2x + (5ix3 + -25i2x2) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + (5ix3 + -25i2x2) + 10x * (-5i + x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + (-5i * 10x + x * 10x) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + (-50ix + 10x2) + -7x2 * (-5i + x) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + -50ix + 10x2 + (-5i * -7x2 + x * -7x2) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + -50ix + 10x2 + (35ix2 + -7x3) + x3(-5i + x)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + -50ix + 10x2 + 35ix2 + -7x3 + (-5i * x3 + x * x3)) fx = (50ix + -250i2 + -35ix2 + 175i2x + 5ix3 + -25i2x2 + -50ix + 10x2 + 35ix2 + -7x3 + (-5ix3 + x4)) Reorder the terms: fx = (50ix + -50ix + -35ix2 + 35ix2 + 5ix3 + -5ix3 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) Combine like terms: 50ix + -50ix = 0 fx = (0 + -35ix2 + 35ix2 + 5ix3 + -5ix3 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) fx = (-35ix2 + 35ix2 + 5ix3 + -5ix3 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) Combine like terms: -35ix2 + 35ix2 = 0 fx = (0 + 5ix3 + -5ix3 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) fx = (5ix3 + -5ix3 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) Combine like terms: 5ix3 + -5ix3 = 0 fx = (0 + -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) fx = (-250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4) Solving fx = -250i2 + 175i2x + -25i2x2 + 10x2 + -7x3 + x4 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Divide each side by 'x'. f = -250i2x-1 + 175i2 + -25i2x + 10x + -7x2 + x3 Simplifying f = -250i2x-1 + 175i2 + -25i2x + 10x + -7x2 + x3 Reorder the terms: f = 175i2 + -250i2x-1 + -25i2x + 10x + -7x2 + x3
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