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(x-2)(x-2i)(x+2i)

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

Factoring and Finding Roots of a Polynomial

This article explores the polynomial (x - 2)(x - 2i)(x + 2i) and its properties.

Understanding the Factors

The polynomial is already in factored form. Let's break down the factors:

  • (x - 2): This factor represents a linear term with a root of x = 2.
  • (x - 2i): This factor represents a linear term with a root of x = 2i, where 'i' is the imaginary unit (i² = -1).
  • (x + 2i): This factor represents a linear term with a root of x = -2i.

Expanding the Polynomial

To get a better understanding of the polynomial's form, let's expand it:

  1. Expand the last two factors: (x - 2i)(x + 2i) = x² - (2i)² = x² + 4

  2. Multiply the result with the first factor: (x - 2)(x² + 4) = x³ - 2x² + 4x - 8

Therefore, the expanded form of the polynomial is x³ - 2x² + 4x - 8.

Finding the Roots

The roots of the polynomial are the values of 'x' that make the polynomial equal to zero. We already know these from the factored form:

  • x = 2
  • x = 2i
  • x = -2i

Key Takeaways

  • Complex Conjugates: The factors (x - 2i) and (x + 2i) are complex conjugates. This is common in polynomials where imaginary roots exist. They always come in conjugate pairs.
  • Polynomial Degree and Roots: The polynomial is a cubic (degree 3) and has three distinct roots. This is consistent with the Fundamental Theorem of Algebra which states that a polynomial of degree 'n' has exactly 'n' roots (counting multiplicity).

This analysis highlights the relationship between factoring a polynomial, its roots, and the importance of understanding complex numbers in algebra.

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