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Polynomial factoring is a critical skill when we are dealing with polynomial functions. It involves expressing a polynomial as the product of its factors, which can be polynomials of lower degrees. Just like the number 6 can be factored into 2 times 3, a polynomial such as \( f(z) = z^2 - 5z + 6 \) can be factored into \( (z - 2)(z - 3) \).

Often, the first step is to find the greatest common factor (GCF) among the terms, as shown in our exercise. Here, 3 was factored out to simplify the polynomial, making the subsequent steps easier. If the polynomial can’t be readily factored, as sometimes happens with higher-degree polynomials, other methods such as the Rational Root Theorem or synthetic division might be more suitable.

The Rational Root Theorem is a powerful tool that can help when tackling tougher polynomials that don't easily factor. It states that the possible rational roots of a polynomial equation \( ax^n + bx^{n-1} + \dots + k = 0 \) are given by the factors of the constant term \( k \) divided by the factors of the leading coefficient \( a \).

Let's say we have a constant term of 6 and a leading coefficient of 2. The possible rational roots would then be \( \pm1 \) , \( \pm2 \) , \( \pm3 \) , and \( \pm6 \), as well as their halves. When we apply this theorem to our exercise, we narrow down the list of potential zeros to test, which streamlines the process of finding the actual zeros.

Once we have a list of potential rational zeros from the Rational Root Theorem, synthetic division provides a way to test these candidates efficiently. Unlike long division, synthetic division is a shortcut that allows us to divide a polynomial by a divisor of the form \( z - c \) quickly and with less work. Done correctly, it will result in a new polynomial of one degree less and possibly a remainder.

If the remainder is zero, congratulations, you've found a root! This root can then be used to factor the polynomial further. In our exercise, synthetic division confirmed that \( z = -1/2 \) and \( z = 1 \) are indeed roots, drastically cutting down the complexity of the problem and bringing us closer to finding all zeros.

The roots of a polynomial function are the solutions to the equation \( f(z) = 0 \) . These values are where the graph of the polynomial intersects the horizontal axis (often the x-axis in graphs). In other words, at the roots, the polynomial has a value of zero. Finding these roots is essential in many areas of mathematics and applied science, as they can reveal critical points of the function such as maximums, minimums, and points of inflection.

The exercise’s final step involves setting each factor equal to zero and solving for \( z \). The factors we identify either through factoring, the Rational Root Theorem, or synthetic division allow us to pinpoint these roots precisely. For the polynomial given in the exercise, the roots are found to be \( z = -1/2 \) , \( z = 1 \) , and \( z = 3/4 \) , which are the crucial points where the function crosses the axis.