Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Reasoning Let \(f\) be a fourth-degree polynomial function with real coefficients. Three of the zeros of \(f\) are \(3,1+i\), and \(1-i\) Explain how you know that the fourth zero must be a real number.

Step-by-Step Solution

Answer

The fourth zero of this fourth-degree polynomial must be a real number because complex roots in a polynomial with real coefficients occur in conjugate pairs. Since we already have a conjugate pair, any remaining zeros are necessarily real.

Step 1: Understanding Polynomial Equations

In a polynomial equation, the degree of the polynomial defines the maximum number of roots it can have. A fourth-degree polynomial can have four roots.

Step 2: Examining the Given Roots

Among the given roots, two roots \(1+i\) and \(1-i\) are complex numbers. It's important to understand that in a polynomial with real coefficients, complex roots always occur in conjugate pairs. A conjugate pair means if \(a+bi\) is a root, then \(a-bi\) will also be a root.

Step 3: Final Deduction

Since \(1+i\) and \(1-i\) are complex conjugates, and \(3\) is the real number that is already a root, the fourth root cannot be a complex number otherwise it would violate the rule of complex roots occurring in pairs for polynomials with real coefficients. Therefore, it must be a real number.

Key Concepts

Complex roots

When dealing with polynomial equations, complex roots play an essential role. A complex root is a solution to a polynomial equation that includes imaginary numbers.
Imaginary numbers are numbers that when squared give a negative result. They are denoted with an 'i', where \(i^2 = -1\). Complex roots typically come in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.
It's crucial for students to recognize these roots, as handling them is a key skill in higher mathematics. Complex roots often arise in quadratic equations when the discriminant (expression under the square root in the quadratic formula) is negative.

• Understanding how complex roots interact with each other is vital in solving polynomial equations.
• Knowing they might not always represent a point you can find on the real number line helps in visualizing polynomial functions.

Conjugate pairs

When a polynomial has real coefficients, complex roots must occur in conjugate pairs.
A conjugate pair consists of two numbers: \(a + bi\) and \(a - bi\). You can easily identify these pairs by looking at the signs between \(b\) and \(i\) — if they are opposite, you have a conjugate pair.
This concept occurs because the coefficients of our polynomial are real numbers. A single complex number introduces an imaginary component which can alter the reality of the equation. Thus, a pair, \((a + bi)\) and \((a - bi)\), is needed to ensure the resulting product is real:

• Polynomials with real coefficients cannot have an odd number of complex roots.
• If a polynomial has a complex root, its conjugate must also be a root.

This ensures that any imaginary components cancel out, resulting in a polynomial with real coefficients.

Real coefficients

Polynomials often feature real coefficients, meaning each coefficient is a real number.
This property is significant because it influences the nature of the polynomial's roots.
For any polynomial with real coefficients, if a complex root exists, its conjugate must also be a root and must appear as a pair. This guideline helps in analyzing the potential roots of a polynomial without performing extensive calculations.

• Real coefficients ensure the polynomial function behaves predictably over real numbers.
• This property also guarantees certain symmetry in the root structure of the polynomial equation.

In the context of this specific polynomial problem, having real coefficients means that with two complex roots that are a conjugate pair, the fourth root must be real to maintain the even count of non-real roots. This maintains balance and conforms with the intrinsic characteristics defined by real coefficients.