(x-1)(x-1)(x+2i)(x-2i)

3 min read Jun 17, 2024

Factoring and Finding Roots of a Polynomial: (x-1)(x-1)(x+2i)(x-2i)

This article explores the polynomial (x-1)(x-1)(x+2i)(x-2i), focusing on its factorization and roots.

Understanding the Factors

The polynomial is already presented in factored form, making it easier to understand its roots. Let's break down each factor:

(x - 1): This factor represents a root at x = 1. Since it appears twice, we know x = 1 is a double root.
(x + 2i): This factor represents a root at x = -2i.
(x - 2i): This factor represents a root at x = 2i.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots, counting multiplicity. In our case, the polynomial has degree 4 (as we have four factors) and we have identified four roots:

x = 1 (multiplicity 2)
x = -2i
x = 2i

Expanding the Polynomial

To visualize the polynomial in its standard form, we can expand it:

(x-1)(x-1)(x+2i)(x-2i) = (x^2 - 2x + 1)(x^2 + 4) = x^4 - 2x^3 + 5x^2 - 8x + 4

This demonstrates how the polynomial's factored form reveals its roots, while the expanded form shows its structure as a sum of terms.

Conclusion

Understanding the factorization of a polynomial allows us to easily determine its roots. The polynomial (x-1)(x-1)(x+2i)(x-2i) has a double root at x = 1 and two complex roots at x = -2i and x = 2i. This knowledge allows for deeper analysis and applications in various mathematical fields.