(3%2B2i)(1-4i).jpeg "(2-i)(3+2i)(1-4i)")
Expanding and Simplifying Complex Expressions: (2-i)(3+2i)(1-4i)
This article explores the process of multiplying and simplifying the complex expression (2-i)(3+2i)(1-4i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as √-1.
Expanding the Expression
We will expand the given expression by applying the distributive property (often referred to as FOIL for binomials).
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Start with the first two factors:
(2-i)(3+2i) = 2(3+2i) - i(3+2i) = 6 + 4i - 3i - 2i²
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Simplify, remembering that i² = -1:
= 6 + 4i - 3i + 2 = 8 + i
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Now multiply the result by the third factor:
(8+i)(1-4i) = 8(1-4i) + i(1-4i) = 8 - 32i + i - 4i²
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Simplify again, using i² = -1:
= 8 - 32i + i + 4 = 12 - 31i
Final Result
Therefore, the simplified form of the expression (2-i)(3+2i)(1-4i) is 12 - 31i.
Key Points
- Complex numbers are numbers that involve the imaginary unit 'i'.
- The distributive property is used to expand complex expressions.
- Remember that i² = -1, which is crucial for simplifying the expressions.