91Ó°ÊÓ

Polynomial functions are mathematical expressions that consist of variables and coefficients, using just the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. Visualize them as mathematical 'machines' that take inputs and produce outputs following a set of algebraic rules. For example, the polynomial function given in our exercise,
\[ f(x)=3x^{3}-19x^{2}+33x-9 \]
is a cubic function because the highest degree of the variable 'x' is three. The 'degree' dictates the function's shape and the number of zero points (roots) it can have. Every polynomial function has coefficients, in this case, 3, -19, 33, -9, which when combined with the variables and their exponents, define the curve's position and characteristics on the graph.

The Factor Theorem emerges as a bridge between algebra and calculus and is incredibly useful when determining the zeros of polynomial functions. It states that a polynomial has a factor of \( x - p \) if and only if \( p \) is a zero of the function. Essentially, if you can find a value for \( x \) that makes the polynomial equal to zero, then \( x - p \) is a factor of that polynomial.

For instance, if you plug \( x = p \) into \( f(x) \) and the result is zero, \( f(p) = 0 \) confirms \( p \) as a root, and you've successfully 'factored' a slice of the polynomial. In our exercise, when \( f(1) = 0 \) and \( f(3) = 0 \) and \( f(1/3) = 0 \) we confirm that \( x - 1 \) and \( x - 3 \) and \( x - 1/3 \) are factors of the polynomial function.

The Rational Root Theorem is a diagnostic tool in algebra that when given a polynomial, allows us to list all possible rational roots that the polynomial might have. It’s akin to having a 'suspects' list in a detective story, except the suspects are potential rational zeros. According to this theorem, any rational zero, expressed in the form \( p/q \) where \( p \) and \( q \) are integers, must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.

Applying this rule, like a mathematical sieve, we check which 'suspects' are indeed zeros of the function. In the exercise provided, by listing all factors of the constant term (-9) and the leading coefficient (3), we get a condensed list of possible zeros to test, which thereby leads us to discover the rational zeros of the polynomial 1, 3, and \( 1/3 \).