Problem 66 - Precalculus : Building Concepts and Connections

Question

Let \(p(x)=x^{5}+x^{3}-2 x\). (a) Show that \(p\) is symmetric with respect to the origin. (b) Find a zero of \(p\) by inspection of the polynomial expression. (c) Use a graphing utility to find the other zeros. (d) How do you know that you have found all the zeros of \(p ?\)

Step-by-Step Solution

Answer

a) The function \( p \) is symmetric with respect to the origin as \( p(-x) = -p(x) \). b) A zero of the polynomial is x = 0 by inspection. c) The other zeros of \( p \) can be found using a graphing utility by identifying the x-intercepts of the plot of the function. d) All the zeros of \( p \) have been found by ensuring the number of zeros, including multiplicities, equals the degree of the polynomial as per the fundamental theorem of algebra.

Step 1: Proving symmetry of function

A function \( f(x) \) is symmetric with respect to the origin if \( f(-x) = -f(x) \) for all x in the function's domain. With \( p(x) = x^{5}+x^{3}-2x \), computing \( p(-x) \) gives \( -x^{5} -x^{3} +2x \), which is -p(x). Therefore, the function is symmetric with respect to the origin.

Step 2: Find a zero of the polynomial by inspection

By inspection, it can be seen that a zero of the polynomial exists at x = 0, as substituting this into the polynomial expression gives \( p(x) = 0^{5}+0^{3}-2(0) = 0 \).

Step 3: Find other zeros using a graphing utility

By graphing the function \( x^{5}+x^{3}-2x \) using a graphing utility, you can find the other zeros of \( p \) by finding the x-intercepts of the graph.

Step 4: Verifying all zeros are found

The fundamental theorem of algebra states that a non-constant polynomial has the number of roots equal to its degree. Since the given polynomial \( p \) is of degree 5, it must have exactly 5 roots. Be sure to account for roots with multiplicity. As found in Step 3, when you graph the function, check that there are exactly five x-intercepts including any repeated roots. Be aware that some roots might be complex numbers.

Key Concepts

Symmetry with Respect to the Origin

Understanding symmetry in mathematical functions is crucial for analyzing graphs and predicting their behavior. Symmetry with respect to the origin occurs when reflecting a function across both the x-axis and the y-axis results in the same graph. Concretely, for a function to exhibit origin symmetry, it has to satisfy the condition that for every input value x, the function value at -x is the negative of the function value at x, formally written as

• If \( f(-x) = -f(x) \), then the function has origin symmetry.

For the polynomial \( p(x) = x^{5}+x^{3}-2x \), when we substitute \( -x \) in place of \( x \), we indeed get \( -p(x) \). This algebraic manipulation proves that the polynomial function \( p \) is symmetric with respect to the origin. This characteristic tells us that for every point \( (x, y) \) on the graph, there is a corresponding point \( (-x, -y) \), mirroring the point across both axes.

Graphing Polynomial Functions

When graphing polynomial functions, it's essential to consider various features, such as intercepts, end behavior, and turning points. A graph provides a visual representation of the solution sets and helps us understand the behavior of the function across its domain.

When you plot the function \( p(x) = x^{5}+x^{3}-2x \), you notice how it crosses or touches the x-axis at its zeros. The x-intercepts correspond to the real zeros of the function. By determining these intercepts, either through graphing utilities or algebraic methods, we can learn not just where the function has zeros but also about its overall shape and structure. This graphing process is invaluable for identifying the number of real roots and their approximate values, which is the starting point for more precise algebraic calculations.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone in understanding polynomial functions. It asserts that

• Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

This implies that the degree of the polynomial indicates the maximum number of roots, including complex roots, that the polynomial can have. For the polynomial \( p(x) = x^{5}+x^{3}-2x \), we'd expect to find five roots since it's a fifth-degree polynomial. Some of these roots might be real numbers, which we can see when the graph intersects the x-axis, and others might be complex, not visible on the standard Cartesian plane but essential for a complete root analysis. This theorem reassures us that our search for the roots is exhaustive and that all possible solutions are accounted for.

Polynomial Root Multiplicity

Multiplicity refers to the number of times a particular root occurs for a polynomial. In other words,

• A root has a multiplicity of 2 if it is a double root, meaning the graph of the polynomial just touches the x-axis at that point and turns around.
• If a root has a multiplicity of 3 or higher, the graph will flatten out more at the intercept as it crosses or touches the x-axis.

Roots with even multiplicities suggest that the polynomial does not cross the x-axis, whereas roots with odd multiplicities do cross the axis at those points. Knowing the multiplicity of a zero is instrumental for both graphing the polynomial accurately and understanding its behavior near the roots. In our polynomial \( p(x) = x^{5}+x^{3}-2x \), we can see at x=0, there's a root with odd multiplicity, since the graph crosses the axis at this point. Paying attention to root multiplicities is key to deciphering the intricacies of polynomial functions.