(x-3)(x-(1+i))(x-(1-i))

Jasonbradley

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Jun 17, 2024

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Factoring and Expanding the Polynomial (x-3)(x-(1+i))(x-(1-i))

This article explores the process of factoring and expanding the polynomial expression: (x-3)(x-(1+i))(x-(1-i)). We will demonstrate how to manipulate this expression to achieve both factored and expanded forms.

Factoring the Polynomial

The given expression is already factored into three linear terms. Each term represents a root of the polynomial:

• x = 3
• x = 1 + i
• x = 1 - i

Here, i represents the imaginary unit, where i² = -1.

Expanding the Polynomial

To expand the polynomial, we can systematically multiply the factors:

1. Multiply the last two factors: (x-(1+i))(x-(1-i)) = x² - (1-i)x - (1+i)x + (1+i)(1-i)

2. Simplify the multiplication: = x² - x + ix - x - ix + 1 + i² = x² - 2x + 1 - 1 (since i² = -1) = x² - 2x

3. Multiply the result by the first factor: (x-3)(x² - 2x) = x³ - 2x² - 3x² + 6x = x³ - 5x² + 6x

Therefore, the expanded form of the polynomial is x³ - 5x² + 6x.

Key Observations

• The original factored form reveals the roots of the polynomial.
• The expanded form represents the polynomial in its standard form.
• The polynomial has one real root (x=3) and two complex conjugate roots (1+i and 1-i).

Understanding the process of factoring and expanding polynomials is fundamental in various mathematical applications, including solving equations, finding roots, and analyzing functions.