Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Use the concepts of this section to answer the following. What are the possible numbers of real zeros (counting multiplicities) for a polynomial function with real coefficients of degree \(5 ?\)

Step-by-Step Solution

Answer

A polynomial of degree 5 can have 1, 3, or 5 real zeros.

Step 1: Identify the Degree of the Polynomial

The exercise states that the polynomial function has a degree of 5. This means the highest power of the variable in the polynomial is 5.

Step 2: Determine Maximum Possible Real Zeros

For any polynomial function of degree \( n \), there can be at most \( n \) real zeros, considering multiplicities. Since our polynomial is of degree 5, it can have at most 5 real zeros.

Step 3: Consider the Possible Number of Real Zeros

Since complex zeros of polynomials with real coefficients come in conjugate pairs, any odd number from 1 to 5 could represent the number of real zeros because the remaining zeros, if any, must be complex and in pairs.

Step 4: Conclude Possible Real Zeros

Therefore, the possible numbers of real zeros (counting multiplicities) for our polynomial of degree 5 are 1, 3, or 5.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the values of the variable that make the polynomial equal to zero. In simpler terms, these are the x-values where the graph of the polynomial crosses or touches the x-axis. Real zeros are crucial because they indicate the roots of the polynomial on a real number line.

• For a polynomial with real coefficients, real zeros can be isolated using synthetic division or the quadratic formula for second-degree polynomials.
• The number of real zeros may include both solutions that cross the x-axis and those that only touch it without crossing. These are counted according to their multiplicities, which means repeated occurrences of the same zero.

While every polynomial with real coefficients has zeros, not all of them are necessarily real; some may be complex instead.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It plays a significant role in determining the behavior and nature of the polynomial function's graph and its zeros.

• The degree tells us the maximum number of zeros a polynomial can have, as well as the maximum number of x-intercepts the graph might show.
• For instance, a polynomial of degree 5 could potentially have up to 5 real zeros.
• Understanding the degree helps predict the end behavior of the graph—it indicates how the polynomial behaves as the variable approaches positive or negative infinity.

Identifying the degree is the first step to solving any polynomial-related problem because it structurally defines the polynomial's fundamental characteristics.

Complex Zeros

Complex zeros occur when a polynomial function does not cross the x-axis at certain points. Instead, these zeros comprise both a real and an imaginary part. They exist in complex conjugate pairs if the polynomial has real coefficients. This distinct feature has helpful implications.

• Since they come in pairs, the presence of complex zeros can affect the number of real zeros a polynomial of certain degree might have.
• In polynomial algebra, these zeros help complete the set of roots, ensuring the polynomial's factorization showcases its full range of solutions.
• Even if a polynomial has even real zeros, the complex zeros guarantee the total number of zeros, including multiplicities equals the degree of the polynomial.

Therefore, understanding complex zeros is essential for solving polynomial functions completely and consistently.

Multiplicities

Multiplicity refers to the number of times a particular zero appears in the polynomial. When a zero repeats multiple times, it affects the graph and the polynomial's behavior around that root.

• If a zero has an odd multiplicity, the graph crosses the x-axis at that point. When the multiplicity is even, the graph only touches the x-axis and turns back.
• The sum of the multiplicities of all zeros for a polynomial must match the degree of the polynomial.
• Multiplicity is critical in determining how many times a certain zero should be considered when computing the total numbers of zeros.

Analyzing multiplicities helps ensure that every zero—whether real or complex—is accounted for correctly, giving a complete picture of the polynomial's solutions.