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The degree of a polynomial is a key concept in understanding rational functions. It tells you the highest power of the variable in the polynomial.

For example, in the expression \(x^4 - 3x^3 - 21x^2 + 43x + 60\), the highest power of \(x\) is 4, so the degree of this polynomial is 4.

The degree helps in many ways including finding the horizontal asymptote of a rational function.

In our given function, the degree of the numerator \(x^4 - 3x^3 - 21x^2 + 43x + 60\) is 4 and the degree of the denominator \(x^4 - 6x^3 + x^2 + 24x - 20\) is also 4.

Knowing both degrees are equal helps us decide the method to find the horizontal asymptote.

The leading coefficient is the number in front of the term with the highest power in a polynomial. It plays a vital role in determining the horizontal asymptote of a rational function.

For the polynomial \(x^4 - 3x^3 - 21x^2 + 43x + 60\), the leading coefficient is 1 because the term with the highest power, \(x^4\), has a coefficient of 1.

Similarly, in the polynomial \(x^4 - 6x^3 + x^2 + 24x - 20\), the leading coefficient is also 1.

When the degrees of the polynomials are the same, you find the horizontal asymptote by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. Here, it's \(\frac{1}{1} = 1\).

So, the horizontal asymptote for this function is \(y = 1\).

A rational function is simply the ratio of two polynomials. For example, in \(f(x) = \frac{x^4 - 3x^3 - 21x^2 + 43x + 60}{x^4 - 6x^3 + x^2 + 24x - 20}\), we have one polynomial in the numerator and another in the denominator.

Understanding rational functions includes examining their behavior at extreme values of \(x\) (both large positive and negative).

This behavior is often described by the horizontal asymptote, which tells us where the function stabilizes as \(x\) becomes very large or very small.

The degrees of the polynomials and the leading coefficients are the keys to finding this asymptote.

In our function, both polynomials have the same degree, and both leading coefficients are 1. This allows us to conclude the horizontal asymptote is \(y = 1\).