Question

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$-4,3 i,-3 i, 2 i,-2 i$$

Short answer

The polynomial is \(x^5 + 8x^4 + 61x^3 + 272x^2 + 576x + 576\). This is one correct answer out of many possible.

Step-by-step solution

Understand the Zeros

The given roots of the polynomial are -4, 3i, -3i, 2i, and -2i. The polynomial must have all of these values as solutions to the equation. Some of these roots are complex numbers, but according to the Fundamental Theorem of Algebra, complex roots always come in conjugate pairs in polynomials that have real coefficients.

Build Polynomial Factors

To build a polynomial from the roots, write factors in the form of (x - root). \[x - (-4),\] \[(x - 3i)(x + 3i),\] \[(x - 2i)(x + 2i)\] The complex roots form quadratic factors with real coefficients upon multiplication.

Simplify the Factors

Simplify the factors obtained in step 2. \[x + 4,\] \[(x^2 - (3i)^2),\] \[(x^2 - (2i)^2)\] Simplify these expressions to get: \[x + 4,\] \[(x^2 + 9),\] \[(x^2 + 4)\]

Construct the Polynomial

To generate the polynomial, multiply the factors together.\[(x+4)(x^2+9)(x^2+4)\] When multiplied out, this gives the following polynomial: \[x^5 + 8x^4 + 61x^3 + 272x^2 + 576x + 576\]

Key concepts

Complex Zeros

When dealing with polynomials with real coefficients, understanding complex zeros is essential. Complex zeros, like those in the form of \(a + bi\), usually appear in pairs known as complex conjugates. If a polynomial has complex zeros and real coefficients, these zeros are paired as \(a + bi\) and \(a - bi\).
This pairing ensures the polynomial's result remains real when evaluated. For example, if \(3i\) is a zero, \(-3i\) must also be a zero. Similarly, the zeros \(2i\) and \(-2i\) are a conjugate pair.
It's crucial to recognize this pattern to correctly construct polynomial factors since both the zero and its conjugate contribute to forming factors that have real number coefficients.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone concept in understanding polynomials. It states that every non-zero, single-variable polynomial with complex coefficients has exactly as many zeros as its degree, including multiplicities. This theorem underscores that a polynomial of degree \(n\) will have \(n\) roots in the complex number system.
For polynomials with real coefficients, the theorem implies that any complex zeros must occur in conjugate pairs, maintaining the polynomial's real nature. Thus, understanding this theorem helps in predicting the number and type of zeros a polynomial will have, assisting in forming the polynomial from given zeros.
In our example, with zeros including \(-4, 3i, -3i, 2i, -2i\), the total count is 5, matching a fifth-degree polynomial as expected.

Polynomial Factors

Polynomial factors are expressions found by writing a polynomial in terms of its zeros. The process begins by using the given zeros to form linear factors, typically in the form of \((x - root)\). Each zero directly translates to a factor, and these factors multiply to form the polynomial.
For example, zeros \(-4\) translates to the factor \((x + 4)\). Complex zeros such as \(3i\) and \(-3i\) combine to form a quadratic factor \((x^2 + 9)\), removing the imaginary components in multiplication. Similarly, zeros \(2i\) and \(-2i\) form \((x^2 + 4)\).
Multiplying these individual factors together will yield the complete polynomial. This understanding is crucial for generating polynomial equations from a set of given zeros, ensuring that the output polynomial has the correct degree and properties consistent with having real coefficients.