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

2 min read Jun 16, 2024
(2+i)(4i^3-3i^2+2i-1)

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

  1. Simplify the powers of 'i':

    • i³ = i² * i = -1 * i = -i
    • i² = -1

  2. Substitute the simplified powers of 'i' into the expression: (2 + i)(4(-i) - 3(-1) + 2i - 1)

  3. Distribute and simplify: (2 + i)(-4i + 3 + 2i - 1) (2 + i)(-2i + 2)

  4. Expand using the distributive property (FOIL): (2 * -2i) + (2 * 2) + (i * -2i) + (i * 2) -4i + 4 - 2i² + 2i

  5. Substitute i² with -1: -4i + 4 - 2(-1) + 2i

  6. 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.

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