Finding the Polynomial of Smallest Degree with Integer Coefficients Given Specific Zeros

Introduction

Understanding how to construct a polynomial with specified zeros, especially when dealing with complex numbers and integer coefficients, is a fundamental concept in algebra. This article will walk you through the process of identifying the polynomial of the smallest degree with integer coefficients that has specific zeros. Specifically, we will focus on the zeros 3, (1 i), and (-1).

Concept of Complex Conjugate Roots

It is crucial to remember that if a polynomial has real coefficients and one of its zeros is a complex number, then the complex conjugate must also be a root. This is because the coefficients of the polynomial are real, and the polynomial equation must be satisfied for the complex roots.

Identifying the Zeros

Given the zeros for the polynomial: 3 1 i 1-i -1

Constructing the Polynomial

The polynomial can be constructed by taking the product of factors corresponding to each root. For each root ( r ), the factor is ( (x - r) ).

Factors for Each Root

For a root of 3, the factor is:

(x - 3)

For the complex root (1 i), its conjugate (1 - i) must also be a root, so the factors are:

(x - (1 i)) and (x - (1 - i))

For the root (-1), the corresponding factor is:

(x 1)

Combining the Factors

Combining all the factors, we get:

(P(x) (x - 3)(x - (1 i))(x - (1 - i))(x 1))

Simplifying the Complex Factors

First, we simplify the product of the complex conjugates:

((x - (1 i))(x - (1 - i)))

This can be expanded as:

((x - 1 - i)(x - 1 i) (x - 1)^2 - (i)^2 (x - 1)^2 1 x^2 - 2x 1 1 x^2 - 2x 2)

Now we have the polynomial as:

(P(x) (x - 3)(x^2 - 2x 2)(x 1))

Expanding and Simplifying the Polynomial

First, expand ( (x^2 - 2x 2)(x 1) ):

((x^2 - 2x 2)(x 1) x^3 - 2x^2 2x x^2 - 2x 2 x^3 - x^2 2)

Now, multiply by ( (x - 3) ):

(P(x) (x - 3)(x^3 - x^2 2) x^4 - x^3 2x - 3x^3 3x^2 - 6 x^4 - 4x^3 3x^2 2x - 6)

Final Result

The polynomial of the smallest degree with integer coefficients that has the zeros 3, (1 i), and (-1) is:

(P(x) x^4 - 4x^3 3x^2 2x - 6)

Conclusion

This step-by-step process demonstrates the method of constructing a polynomial with specific integer coefficients and complex zeros. Understanding these steps is crucial for anyone studying algebra and polynomial theory.