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In mathematics, a zero of a function, also known as a root, is any value of the variable that makes the function equal to zero. To determine if a value is a zero of a polynomial function, you substitute the value into the polynomial. If the result is zero, then the value is indeed a zero, or a root, of the polynomial function.

For example, if you are given a polynomial function, say \( f(x) = 3x^5 - 2x^4 + x^3 - 16x^2 + 3x - 8 \), and you want to find out if \( x = \frac{1}{7} \) is a zero of the function, you would substitute \( \frac{1}{7} \) for \( x \) in the expression and simplify it.

If the final result is 0, then \( \frac{1}{7} \) is a zero of this polynomial function; if not, \( \frac{1}{7} \) is not a zero.

Polynomial evaluation involves substituting a particular value for the variable in a polynomial and then simplifying the expression to arrive at a numerical result. This process is fundamental in determining various properties of polynomials, such as finding its zeros or calculating it at specific points.

Let's break down the process of evaluating the polynomial \( f(x) = 3x^5 - 2x^4 + x^3 - 16x^2 + 3x - 8 \) at \( x = \frac{1}{7} \):

• First, replace every instance of \( x \) in the polynomial with \( \frac{1}{7} \).
• Next, calculate each term by raising \( \frac{1}{7} \) to the respective powers and then multiplying by the coefficient.
• Finally, add up all these terms to get the result.

If the evaluated result is zero, that means \( \frac{1}{7} \) would be a root of the function.

Rational zeros refer to zeros of a polynomial that can be expressed as a fraction of two integers. The Rational Zero Theorem helps in identifying possible rational zeros of polynomial functions. According to the theorem, if a polynomial has a rational zero \( \frac{p}{q} \), then \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.

For instance, consider a polynomial such as \( f(x) = 3x^5 - 2x^4 + x^3 - 16x^2 + 3x - 8 \). Since the constant term is -8 and the leading coefficient is 3, by applying the Rational Zero Theorem, we look for potential rational zeros among the fractions formed by the factors of -8 and 3.

• Identify all factors of -8 (\( \pm 1, \pm 2, \pm 4, \pm 8 \)).
• Identify all factors of 3 (\( \pm 1, \pm 3 \)).
• Form possible rational zeros (\( \frac{p}{q} \)).

This gives us potential candidates for rational zeros, which need to be tested to determine if they are actual zeros of the polynomial.