How to Construct a Polynomial Function Given Specific Zeros: -2i, 2i, and √2

When working with polynomial functions, understanding how to construct a polynomial based on given zeros (roots) is a fundamental skill. This article will guide you through the process step-by-step, explaining how to create a polynomial function with the zeros -2i, 2i, and √2. By the end of this article, you will have a clear understanding of the construction process and the logic behind it.

About Zeros and Factors

Zeros (roots) of a polynomial are the values of x that make the polynomial equal to zero. A polynomial with zeros -2i, 2i, and √2 can be expressed as a product of factors derived from these zeros. Known as the Factor Theorem, if x a is a zero, then (x - a) is a factor of the polynomial.

Step-by-Step Construction of the Polynomial

To construct a polynomial with the given zeros, follow these steps:

Identify the Roots: The given roots are -2i, 2i, and √2. Form Factors: For each root, write the corresponding factor. For x -2i, the factor is x 2i; for x 2i, the factor is x - 2i; and for x √2, the factor is x - √2. Multiply the Factors for Complex Roots: Since complex roots occur in conjugate pairs, we already have one factor for each complex root. Now, multiply these complex factors: (x 2i)(x - 2i). Combine with the Real Root Factor: After multiplying the complex factors, multiply the result by the factor corresponding to the real root, i.e., x - √2. Expand the Polynomial: Finally, expand the resulting polynomial to put it in standard form. Final Polynomial: The final polynomial is in the standard form and has the given zeros.

The Polynomial Derivation

Let's go through the derivation step-by-step:

Form the factors for the given roots: (x 2i) (x - 2i) (x - √2) Multiply the complex factors:

(x 2i)(x - 2i) x^2 - (2i)^2 x^2 - 4(-1) x^2 4

Combine with the real root factor:

x^2 4 (x - √2) (x^2 4)(x - √2)

Expand the polynomial:

(x^2 4)(x - √2) x^3 - √2x^2 4x - 4√2

Final Polynomial:

Therefore, the polynomial function is:

fx x^3 - √2x^2 - 4x - 4√2

This polynomial has the specified zeros: -2i, 2i, and √2.

Polynomial Function with Integer Coefficients

For polynomials with integer coefficients, complex roots must also have their conjugates. The conjugate of -√2 is √2. Therefore, we can include √2 and -√2 as roots to ensure integer coefficients:

Include the conjugate √2 and -√2.

Form the polynomial: (x - √2)(x √2)(x - 2i)(x 2i).

Simplify: (x^2 - 2)(x^2 4).

Expand: x^4 4x^2 - 2x^2 - 8 x^4 2x^2 - 8.

This polynomial is monic (leading coefficient 1), has the smallest degree, and has integer coefficients. The final polynomial with integer coefficients is:

x^4 2x^2 - 8