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**Fifth Degree Equation
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**By**

**Jane Yun
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**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 ****Ō***a***
Õ, investigate the shape of graph for
different values of Ō***a*** Õ****, and look for relationships. Using a
Graphing Calculator, we will start the exploration of **

*x*^{5} = (*x*^{2} - *y*^{2})(*x*^{2 }– *ay*^{2})

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

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**Notice ****that as Ō a Ō
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**

**When
a = 0, the negative x**

**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 ŌaÕ increases.**

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**LetÕs
try when a = -1**

** The shape of graph
changed to more like a HersheyÕs Chocolate, but ****the
intercepts of graph are at**** points (0, 0) and (1,** **0)**** and the domain of graph is (****°****, 1].
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Then, letÕs investigate when

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**LetÕs
look when we put all the graphs together.
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**Based
on the results from the graphs, we can draw few conclusions about the graph of ***x*^{5}** = ( x^{2} - y^{2})(x^{2 }–
ay^{2})**

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

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