Problem 78 - College Algebra

Question

Write equations for several 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

Odd degree polynomial functions will always have at least one real zero since they will always cross the x-axis. However, even degree polynomial functions may or may not have a real zero - they can be constructed in a way that they do not cross the x-axis at all.

Step 1: Write and Graph Odd Degree Polynomial Functions

To begin, write a few examples of odd-degree polynomials, such as \(y = x^3\), \(y = -x^3 + 2x\), and \(y = x^5 - 4x^3 + 2x\). Plot these functions on a graph, and it quickly becomes clear that for each value of x, you have a corresponding y value. Therefore, these functions each cross the x-axis at least once.

Step 2: Analyze Odd Degree Polynomial Functions

Now it's time to analyze the trend of these odd-degree functions graphically. Does it seem possible for these functions to have no real zeros? Odd degree polynomial functions will always start and finish at opposite sides of the y-axis, meaning there is at least one real zero in every case.

Step 3: Write and Graph Even Degree Polynomial Functions

Next, write a few examples of even-degree polynomials like, \(y = x^2\), \(y = -x^4 + 2x^2\), and \(y = x^6 - 4x^4 + 2x^2\). Plot these functions, and you can observe that the graph always starts and finishes on the same side of the y-axis, whether it's from positive to positive or negative to negative side.

Step 4: Analyze Even Degree Polynomial Functions

Take some time analyzing the plot of even degree polynomials. As it's observed in their graphs, it's entirely possible for a polynomial function of even degree to not cross the x-axis at all, specifically when all coefficients of the polynomial increase the function's value, such as the polynomial \(y=x^4+3x+3\). Hence, polynomial functions of even degrees can have no real zeros unlike ones of odd degrees.