Fifth Degree Equation

Jane Yun

In early nineteenth century, Ruffini and Abel showed that the equations of the fifth degree could not be solved with radicals. The general quinitic is thus not like general quadratic, cubic, or biquadratic. Nevertheless, it presents a problem in algebra, which theoretically can be solved by algebraic operations. Only, these operations are so hard that they cannot be expressed by the symbols of radicals.

In this exploration, we will explore the following equation by changing the values of ‘a ’, investigate the shape of graph for different values of ‘a ’, and look for relationships. Using a Graphing Calculator, we will start the exploration of

x⁵ = (x² - y²)(x²– ay²)

First, by inspecting the equation, I came into the conclusion that when x = 0, y = 0, and when y=0, x =1. Hence, I’m predicting that the intercepts of the graph are the points (0, 0) and (1, 0) and the shape of graph will change as ‘a ’ changes.

Now, let’s begin exploration when a = 0, a = 1, a = 3, a = 5, and a = 10.

Notice that as ‘a ‘ changes from 0 to 10, the intercepts of graph are at points (0,0) and (1, 0), and a graph of equation is symmetric with respect to x-axis. Also notice the change in the shape and the domain of the graph when a = 0, a = 1, a = 3, a = 5, & a = 10, but the range of graph is not changing. Ths range is (-∞, ∞).

When a = 0, the negative x-coordinate side shows the continuously stretching loops, and the positive x-coordinate side shows that the loop gets wider as 'a' decreases to 0.

In the negative x-coordinate side, the graph shows two stretched loops. The loop stretches longer and the width of loop gets smaller in the positive x-coordinate side as ‘a’ increases.

Let’s try when a = -1.

The shape of graph changed to more like a Hershey’s Chocolate, but the intercepts of graph are at points (0, 0) and (1, 0) and the domain of graph is (∞, 1].

Then, let’s investigate when a = -3, a = -5, and a = -10. Notice the change in the shape of graph. The shape of graph changes as 'a' decreases, but the intercept points do not change. Also the domain changes as 'a' changes, and the graph is symmetric with respect to the x-axis. Also notice that the graph starts to show a heart shape when a = -1, and the shape of heart stretches as 'a' decreases but till keeping the same intercept points.

Now, let's explore when a = 0.5 & a= -0.5. Notice that the -0.5 loop is wider than the 0.5 loops, and the intercept points did not change. The graph is also symmetric with respect to the x-axis.

Let’s look when we put all the graphs together.

Based on the results from the graphs, we can draw few conclusions about the graph of x⁵ = (x² - y²)(x²– ay²). The intercepts are at points (0, 0) & (0, 1), and the original equation is always symmetric with respect to the x-axis.

The animation is shown below when 'a' changes from -10 to 10.

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