Quartic Polynomial Analysis

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In 1540, a man by the name of Lodovico Ferrari, please be aware that I don’t think his name has anything to do with the sports car, was an Italian mathematician known for discovering the solutions to quartic functions. A quartic function is a function of the form ax^4 + bx^3 +cx^2 +dx+e, where a is a nonzero, which is defined by a polynomial raised to the fourth degree, called quartic polynomial. We will probably go more in depth about these quartic polynomials soon in class. My quartic polynomial was 3x^4 -7x^3 -3x^2 +17x+10, and in this project, I was asked to analyze this polynomial.

Finding the end behavior was one of the first steps of this analysis. The end behavior refers to where the tail ends of the graph are pointing on both the
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The zeros are where the graph crosses the X axis. These zeros are also known as the solutions or the roots. The types of zeros that functions can have are rational zeros, irrational zeros, or complex zeros. Rational zeros are just the basic integers. Irrational zeros have radicals that can’t be removed. Complex zeros have imaginary numbers. Out of these 3 types of zeros, a quartic function can only have 4 rational/irrational zeros, 2 rational/irrational zeros and 2 complex zeros, or 4 complex zeros. I was able to find how many positive and negative zeros I had using the Descartes Rule of Signs. Using the original equation, I counted from left to right, the amount of sign changes, and that gave me number of positive real zeros. I discovered 2 positive and 2 negative real zeros using this method. Now to find the negative number, the exact same process is repeated, only this time, the –X value is substituted in. The next step would ultimately be to use Interval Value Theorem to find what intervals the roots are in. However, I don’t have any sign changes in my table, which is what you would look for to determine the intervals, so I pretty much don’t have any intervals. I think that it may have to do with the fact that it is a parabola, and its specific end behavior. One tail end is going to negative infinity, so all the values are negative and the other tail end is going to positive infinity, so the values are positive. If the graph were to keep going there would never be any sign changes, so there can’t be any intervals. Since there aren’t any intervals, we can move onto the next step of using the Rational Zero Theorem. In order to use this theorem, you would take the P values which are the factors of the constant in the function, and take the q values which are the factors of the coefficient. Then, take each P value and divide it by q, and these will become the possible zeros. These possible zeros

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