(2-i)(3+2i)(1-4i)

2 min read Jun 16, 2024
(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).

  1. Start with the first two factors:

    (2-i)(3+2i) = 2(3+2i) - i(3+2i) = 6 + 4i - 3i - 2i²

  2. Simplify, remembering that i² = -1:

    = 6 + 4i - 3i + 2 = 8 + i

  3. Now multiply the result by the third factor:

    (8+i)(1-4i) = 8(1-4i) + i(1-4i) = 8 - 32i + i - 4i²

  4. 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.

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