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Simplifying Complex Number Expressions: (2 + i)(4i³ - 3i² + 2i - 1)
This article will guide you through the process of simplifying the complex number expression (2 + i)(4i³ - 3i² + 2i - 1).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as √-1.
Simplifying the Expression
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Simplify the powers of 'i':
- i³ = i² * i = -1 * i = -i
- i² = -1
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Substitute the simplified powers of 'i' into the expression: (2 + i)(4(-i) - 3(-1) + 2i - 1)
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Distribute and simplify: (2 + i)(-4i + 3 + 2i - 1) (2 + i)(-2i + 2)
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Expand using the distributive property (FOIL): (2 * -2i) + (2 * 2) + (i * -2i) + (i * 2) -4i + 4 - 2i² + 2i
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Substitute i² with -1: -4i + 4 - 2(-1) + 2i
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Combine like terms: -2i + 6
Final Result
Therefore, the simplified form of the complex number expression (2 + i)(4i³ - 3i² + 2i - 1) is -2i + 6.