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Polynomial division is a technique used to divide polynomials, similar to long division with numbers. The method involves dividing a polynomial (dividend) by another polynomial (divisor) to find the quotient and the remainder. It is highly useful in simplifying polynomials and finding zeros or roots.

When dividing polynomials, we focus on the highest degree term of the dividend and align the divisor underneath. The primary goal is to match or exceed the power of the leading term of the divisor so that subtraction simplifies the expression. Continue this process until the degree of the remaining polynomial is less than the degree of the divisor, which indicates that division is complete.

In the context of finding rational zeros, polynomial division helps verify potential roots. If dividing by a possible root results in a remainder of zero, it indicates that the root is valid. This method is particularly useful after generating possible rational roots using the Rational Root Theorem.

Synthetic division is a simplified form of polynomial division. It's mainly used when dividing by a linear factor of the form \(x - c\). Unlike traditional long division, synthetic division is faster and less cumbersome.

To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Place the potential zero on the left.
• Begin by bringing the first coefficient down as it is.
• Multiply it by the potential zero, and add the result to the next coefficient.
• Continue this process row by row until you reach the end.

Upon finishing, if the last value (remainder) is zero, it confirms the potential zero is indeed a root of the polynomial.

This method is practical for testing many potential rational zeros quickly, allowing you to identify actual roots efficiently. It complements the Rational Root Theorem in narrowing down possible solutions.

Rational zeros of a polynomial are the roots that can be expressed as a fraction \( \frac{p}{q} \). According to the Rational Root Theorem, these potential zeros are derived from factors of the constant term (numerator \(p\)) and factors of the leading coefficient (denominator \(q\)).

To apply this theorem, identify all factors of the constant term and leading coefficient, then form all possible \( \frac{p}{q} \) combinations. These combinations are potential rational zeros.

Once you have the list of possibilities, verify each using synthetic division to check if it is indeed a root. If substituting a potential zero into the polynomial gives a remainder of zero, it means it's a valid rational zero.

Finding rational zeros is crucial as they can help factor the polynomial further, solve algebraic equations, and understand the behavior of polynomial functions.