Problem 42 - Precalculus Mathematics for Calculus

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(S\) has degree 4 and zeros \(2 i\) and \(3 i\).

Step-by-Step Solution

Answer

The polynomial is \( S(x) = x^4 + 13x^2 + 36 \).

Step 1: Identify Given Information

The polynomial, denoted as \( S(x) \), has a degree of 4 and has zeros \( 2i \) and \( 3i \). Since polynomial coefficients must be real, their complex conjugates, \(-2i\) and \(-3i\), must also be zeros of \( S(x) \). Thus, the zeros of \( S(x) \) are \( 2i, -2i, 3i, \) and \(-3i\).

Step 2: Write Down Factors

Using the zeros, express \( S(x) \) as a product of its factors. Each pair of conjugate zeros can be written as a quadratic factor:1. \( (x - 2i)(x + 2i) = x^2 + 4 \).2. \( (x - 3i)(x + 3i) = x^2 + 9 \).

Step 3: Multiply the Factors

Multiply the two quadratic factors to obtain the polynomial:\[ S(x) = (x^2 + 4)(x^2 + 9) \].

Step 4: Expand the Product

Expand the expression:\[ (x^2 + 4)(x^2 + 9) = x^4 + 9x^2 + 4x^2 + 36 = x^4 + 13x^2 + 36 \].

Step 5: Verify the Polynomial

Verify that the polynomial is of degree 4 and that all terms have integer coefficients. In this case, it is indeed a fourth-degree polynomial with integer coefficients.

Key Concepts

Complex Roots

Complex roots are solutions to polynomial equations that include imaginary numbers. Imaginary numbers involve the square root of negative one, denoted as \( i \), with properties such as \( i^2 = -1 \). When a polynomial has real coefficients and a complex root, its complex conjugate must also be a root. For example, if \(2i\) is a root, then \(-2i\) is also a root.
This behavior ensures that any non-real solutions occur in conjugate pairs, maintaining the real nature of polynomial coefficients. This happens because imaginary components cancel out in the quadratic form \((x - ai)(x + ai)\), simplifying to a real polynomial \(x^2 + a^2\).
Understanding complex roots and their conjugates is crucial for forming real polynomials and solving equations involving imaginary numbers.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable within the function when it is expressed in its expanded standard form. It tells us how many solutions or roots the polynomial could potentially have, taking into account both real and complex numbers.
For example, if the polynomial degree is 4, like in the exercise, it can have up to four roots. These roots can be real or come in complex conjugate pairs. In our case, the polynomial has two pairs of complex conjugates: \(2i, -2i, 3i, -3i \). This ensures that the original polynomial is of degree 4.
Knowing the degree helps to predict the behavior of the polynomial graph, including identifying the number of roots.

Conjugate Zeros

Conjugate zeros are pairs of complex numbers that are solutions to polynomial equations with real coefficients. If \( a + bi \) is a zero of the polynomial, then its conjugate \( a - bi \) must also be a zero. This occurs to ensure that the polynomial retains real coefficients, which are often a requirement in practical applications.
To form the polynomial based on these zeros, each conjugate pair can be expressed as a quadratic factor. For instance, with roots \(2i\) and \(-2i\), the quadratic factor becomes \(x^2 + 4\), ensuring all coefficients are integers.
Identifying conjugate zeros is essential for constructing polynomials, as seen in the transformation of roots into quadratic factors. This process simplifies complex roots and results in an equation that aligns with the polynomial's degree and conditions.