Problem 50 - A Graphical Approach to College Algebra

Question

Use the concepts of this section. Explain why a polynomial function of degree 4 with real coefficients has either zero, two, or four real zeros (counting multiplicities).

Step-by-Step Solution

Answer

A degree 4 polynomial with real coefficients has zero, two, or four real zeros due to complex roots occurring in conjugate pairs.

Step 1: Understand the Degree of the Polynomial

A polynomial function of degree 4 can be expressed in the general form: \[ f(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \] where \(a_4 eq 0\). The degree, which is 4, indicates the highest power of \(x\). This implies that the polynomial can have up to 4 zeros, as the Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has \(n\) roots, including complex and repeated roots.

Step 2: Consider Real vs Complex Zeros

Real coefficients in a polynomial imply that complex zeros, if any, occur in conjugate pairs. For a polynomial of degree 4, if it has a complex zero (say \(a + bi\)), its conjugate \(a - bi\) must also be a zero of the polynomial. Hence, the zeros must appear as pairs of either both real or both complex.

Step 3: Analyze the Zeros as Multiplicities

Zeros of a polynomial can be repeated, known as multiplicities. Since the polynomial is degree 4, the total multiplicity of all zeros must equal 4. This means we could have one zero with multiplicity 4, two zeros each with multiplicity 2, or four distinct real zeros. These account for all possible scenarios using real numbers, possible repeated (multiplicities).

Step 4: Examine Every Possibility for Real Zeros

Given the rules above, a polynomial of degree 4 can either have: 1. Zero real zeros if all roots are complex and occur in conjugate pairs (e.g., two pairs of complex conjugates). 2. Two real zeros if there are one complex conjugate pair and two real zeros (with multiplicity types considered). 3. Four real zeros if all roots are real (they may have multiplicities as long as they add up to four). Given these constraints and rules, this is why the options are zero, two, or four real zeros.

Key Concepts

Understanding the Degree of a Polynomial

The degree of a polynomial is directly linked to the number of roots it can have. For instance, when we have a polynomial like \[ f(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \]where \(a_4 eq 0\), this polynomial is said to have a degree of 4. This degree indicates the highest power of \(x\) in the polynomial.
Thanks to the Fundamental Theorem of Algebra, we know that a polynomial of degree \(n\) will have exactly \(n\) roots. This includes both complex and repeated roots.
Hence, a polynomial of degree 4 potentially has 4 roots. These roots might not all be real, but the sum must always reach the polynomial's degree.

The Nature of Complex Zeros

When dealing with polynomial functions with real coefficients, a fascinating concept surfaces: complex zeros. These zeros are particularly interesting because if one exists, it will always have a sibling of sorts, known as its conjugate.
Imagine encountering a complex zero \(a + bi\). In such cases, \(a - bi\) will also appear as a zero of the polynomial. This occurrence comes from the nature of real coefficients, ensuring that complex zeros occur as conjugate pairs.
In a polynomial of degree 4, the complex zeros must therefore balance out in pairs, either all complex or all real. Thus, for a polynomial of degree 4, you might observe:

• Both zeros real
• Both zeros complex

As we explore these scenarios, the real vs. complex balance determines how zeros are counted and structured in polynomial functions.

Multiplicity of Roots in a Polynomial

Multiplicity refers to how many times a particular zero appears in the polynomial's factorization. In a polynomial of degree 4, such as the one we're discussing, all zeros combined must have a total multiplicity of 4.
For instance, a zero with multiplicity 2 appears twice in the factorization. This aspect becomes crucial in understanding how zeros may be distributed.
Consider these configurations for a degree 4 polynomial:

• One zero repeated four times (multiplicity of 4)
• Two distinct zeros, each with a multiplicity of 2
• Four distinct zeros, each with a multiplicity of 1

These scenarios illustrate how the roots of a polynomial can be mapped out, contributing to either real zeros or complex zeros, depending on the arrangement.