Problem 62 - College Algebra

Question

Determine a number of polynomial functions of odd degree and graph each function. Is it possible for the graph to have no real zeros? Explain. Try doing the same thing for polynomial functions of even degree. Now is it possible to have no real zeros?

Step-by-Step Solution

Answer

Polynomial functions of odd degree will always cross the x-axis at least once, therefore, have at least one real zero because they can take negative and positive values. Examples include \(y=x^3\) and \(y=x^5\). On the other hand, polynomial functions of even degree might not cross the x-axis at all if the function does not take both negative and positive values. Therefore, even functions like \(y=(x^2+1)\) and \(y=(x^4+1)\) may not have real zeros.

Step 1: Identify Polynomial Functions of Odd Degree

Let's consider a few polynomial functions of odd degree such as \(y=x^3\) and \(y=x^5\). Polynomial functions of odd degree will always cross the x-axis at least once.

Step 2: Graph Polynomial Functions of Odd Degree

When graphing the functions \(y=x^3\) and \(y=x^5\), it can be observed that they cross the x-axis. This is because the multiplicity of the roots are odd, causing the graph to cross the x-axis.

Step 3: Explain why Polynomial Functions of Odd Degree cannot have no Real Zeros

It is due to the Fundamental Theorem of Algebra which states that every non-constant polynomial function has at least one real or complex zero. Since odd functions can take negative and positive values, they will always have a real root.

Step 4: Identify Polynomial Functions of Even Degree

Let's consider a few polynomial functions of even degree such as \(y=x^2\) and \(y=x^4\). Polynomial functions of even degree could have no real zeros.

Step 5: Graph Polynomial Functions of Even Degree

When graphing the functions \(y=x^2\) and \(y=x^4\), it can be observed that they touch or cross the x-axis. However, there are situations where the even polynomial functions do not have real zeros such as \(y=(x^2+1)\) and \(y=(x^4+1)\).

Step 6: Explain why Polynomial Functions of Even Degree can have no Real Zeros

Even functions always take positive values (or zero) or are always negative. So, they might not cross the x-axis at all if the function does not take both negative and positive values. Therefore, even functions may not have real zeros.

Key Concepts

Odd Degree Polynomial

An odd degree polynomial is a function whose highest exponent on the variable is an odd number. For example, functions like y = x^3 or y = x^5 are odd degree polynomials. These functions are known for their distinct characteristic of crossing the x-axis at least once. This behavior is due to the nature of odd functions, which implies they must take on both positive and negative values, guaranteeing at least one real zero. Understanding this concept is crucial for recognizing how the graph of an odd degree polynomial will behave. In essence, it will always pass through the x-axis, thus reflecting at least one point where the function equals zero.

Graphing Polynomials

Graphing polynomials involves plotting the behavior of the polynomial function on a coordinate plane to visualize its zeros, end behavior, and the nature of its turning points. When graphing, it's essential to identify the degree and leading coefficient, as these will guide the polynomial's end behavior. For odd degree polynomials, the ends of the graph will always be off in opposite directions, while for even degree polynomials, the ends will either both rise or both fall. Tools such as the sign analysis of the polynomial can help predict the regions where the graph will be above or below the x-axis.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem guarantees that polynomial equations will have as many roots as its degree indicates, albeit some or all may be complex numbers. Consequently, an odd degree polynomial must have at least one real root since odd functions must cross the x-axis owing to the fact they reach both positive and negative values. This theorem is a core principle in understanding the nature of polynomials and underlies the reasoning behind graphing and solving them.

Real Zeros of Polynomials

Real zeros of polynomials are the points at which the graph of a polynomial intersects the x-axis. These zeros are crucial when solving polynomial equations since they represent the values where the polynomial is equal to zero. Odd degree polynomials are guaranteed to have at least one real zero due to their crossing nature. Even degree polynomials, however, might not intersect the x-axis at all, therefore showing no real zeros, particularly in cases where the polynomial doesn't take negative values, like y = (x^2 + 1). Identifying real zeros can serve as a first step in sketching a graph and in fully understanding the polynomial's behavior.

Even Degree Polynomial

An even degree polynomial displays quite a different behavior compared to its odd degree counterparts. Functions like y = x^2 or y = x^4 are examples of even degree polynomials. These functions have graphs that either touch or cross the x-axis at their zeros. However, when the function does not change signs, as seen with y = (x^2 + 1) which is always positive, there can be no real zeros. This allows the possibility of an even degree polynomial not intersecting the x-axis, depending on its specific coefficients and constant terms. Recognizing that an even degree polynomial can have no real zeros broadens the understanding of possible graph shapes for polynomial functions.