Jasonbradley
Home

(x-1)(x-5i)(x+5i)

2 min read Jun 17, 2024
(x-1)(x-5i)(x+5i)

Factoring and Expanding the Polynomial (x-1)(x-5i)(x+5i)

This expression involves complex numbers and showcases the concept of complex conjugates. Let's break down its factorization and expansion:

Understanding Complex Conjugates

The key to understanding this expression lies in recognizing that (x-5i) and (x+5i) are complex conjugates.

  • Complex Conjugates: Two complex numbers are complex conjugates if they have the same real part but opposite imaginary parts. The product of complex conjugates is always a real number.

Expanding the Expression

  1. Focus on the Conjugates: Let's begin by expanding (x-5i)(x+5i):

    (x-5i)(x+5i) = x² + 5xi - 5xi - 25i²

    Since i² = -1, we can simplify:

    x² + 5xi - 5xi - 25i² = x² + 25

  2. Final Expansion: Now, we multiply the result by (x-1):

    (x-1)(x² + 25) = x³ - x² + 25x - 25

The Final Result

The fully expanded form of (x-1)(x-5i)(x+5i) is x³ - x² + 25x - 25.

Key Takeaways

  • The expression demonstrates the property of complex conjugates, where the product results in a real number.
  • Expanding the expression involves careful multiplication and simplification, using the knowledge that i² = -1.
  • This example showcases how complex numbers can be used in polynomial expressions.

Related Post

Featured Posts