There is an inconsistency we identified through different reputable websites and textbooks about the degree of a polynomial. If there is an equation 3x^4 + x^3+ 5x^2, then would the degree be 4 or quartic? What is ultimately the difference between the the numerical value of the degree (4) and the type of polynomial (quartic)?

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The function y(x) = 3x^4 + x^3+ 5x^2== a x^4 + b x^3+ c x^2 + d x^1 + e, where a=3, b=1, c=5, d=0, e=0, is of degree 4.

Whether it is quartic depends on your definition, which seems to be not uniquely defined on websites. Here they talk about more than one definition:

While some authors (Beyer 1987b, p. 34) use the term "biquadratic equation" as a synonym for quartic equationhttp://mathworld.wolfram.com/QuarticEquation.html

>> difference between the numerical value of the degree (4) and the type of polynomial (quartic)?

A function can map a numerical value to another value. I'll guess that you are considering both the input and output as real numbers. Since you are talking about polynomials, and since a quartic is a polynomial, then based on a particular definition of quartic, you still end up with a polynomial of degree 4. So, take those coefficients, a, b, c, d, and e and you get a 4th degree polynomial function (i.e., a does not equal 0). No matter whether this function is a quartic polynomial or not (depending upon some author's definition), the function only yields one value for a given value of x, and that value is completely determined by the 5 coefficients, a, b, c, d, and e.

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