Introduction to Rational Functions

A rational function is the quotient of two polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The simplest form of a polynomial is a constant function, such as fx 1, which is indeed a rational function. Let's delve into the details of how fx 1 can be expressed as a rational function.

Understanding Rational Functions

A rational function is defined as any function that can be written as the quotient of two polynomials, where the denominator is not the zero polynomial. In other words, a rational function is any function of the form:

fx P(x) / Q(x)

where P(x) and Q(x) are both polynomials and Q(x) ≠ 0.

Expressing 1 as a Rational Function

The constant function fx 1 can be expressed as the quotient of two identical polynomials. In fact, we can say:

fx x^21 / x^21

This equality holds true for all x ≠ 0. The polynomial x^21 in the numerator and denominator are the same, giving us the constant value 1 for all non-zero x.

The Mathematics Behind

Let's break down why x^21 / x^21 1 for all x ≠ 0:

Consider the polynomial P(x) x^21. When we divide P(x) by itself, we get:

P(x) / P(x) x^21 / x^21

This is valid for all x ≠ 0, as division by zero is undefined. Therefore, the expression simplifies to 1 for all non-zero x.

Generalizing to Other Polynomials

Interestingly, the same concept can be applied to any degree of polynomial. For example:

In each case, the polynomial in the numerator and denominator are identical, resulting in the function fx 1 for all x except the roots of the polynomial (where the polynomial equals zero).

Conclusion

In summary, a rational function can equal 1 by being the quotient of the same polynomial in the numerator and the denominator. The simplest case is the constant function fx 1, which can be written as x^21 / x^21. This property holds true for any polynomial of any degree, as long as the polynomial is not the zero polynomial. Understanding this concept is crucial for advanced studies in algebra and calculus.

If you have any questions or need further clarification, feel free to reach out. Happy learning!