Constructing a Polynomial with Specified Integer Coefficients and Complex Roots

Determining a polynomial equation with specific integer coefficients and complex roots can be an intriguing yet rewarding task in the realm of algebra. This article will guide you through the process of constructing a polynomial with roots of 3i, -3i, and -4. This involves understanding the role of complex conjugates and the steps to simplify and expand polynomial expressions.

Understanding Complex Roots and Their Conjugates

In polynomial equations with real coefficients, if a complex number is a root, its conjugate must also be a root. For example, if 3i is a root, then -3i must also be a root. This property is crucial for constructing a polynomial with integer coefficients.

Step-by-Step Process for Constructing the Polynomial

Let's walk through the process step-by-step:

Step 1: Writing the Factors for Each Root

To start, we write the factors corresponding to each root:

3i contributes the factor x - 3i -3i contributes the factor x 3i -4 contributes the factor x 4

The presence of -3i as a root ensures that its conjugate 3i is automatically included, which is a straightforward application of the properties of complex numbers in polynomials with real coefficients.

Step 2: Constructing the Polynomial

The next step is to construct the polynomial by multiplying these factors together:

P(x) (x - 3i)(x 3i)(x 4)

Step 3: Simplifying the Product of the Complex Factors

Let's simplify the product of the complex factors first:

(x - 3i)(x 3i) x^2 - (3i)^2 x^2 9

This step utilizes the difference of squares formula and the property that (i^2 -1).

Step 4: Multiplying by the Remaining Factor

Now, we need to multiply the result by the remaining factor:

P(x) (x^2 9)(x 4)

Step 5: Expanding the Polynomial

To get the polynomial in standard form, we expand the expression:

P(x) x^3 4x^2 9x 36

This expansion involves distributing each term in one factor by each term in the other factor.

Final Result

Therefore, the final polynomial equation with integer coefficients that has the roots 3i, -3i, and -4 is:

P(x) x^3 4x^2 9x 36 0

This polynomial satisfies the given conditions, demonstrating the step-by-step process of constructing a polynomial with specified roots and integer coefficients.