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A monomial is an algebraic expression that consists of a single term. This term can include numbers, variables, or both. Monomials are a fundamental part of polynomials, but unlike polynomials, they don't include addition or subtraction between terms. Instead, they stand alone as individual expressions.In our example, we have the monomials: \(10x^3yz^3\), \(20x^2z^5\), and \(45xz^3\). When working with monomials, you often analyze their components separately. This means considering the coefficients (the numeric parts) and each variable raised to some power.Understanding monomials is crucial for finding the greatest common factor (GCF) because you break down both numbers and variables to find commonalities. With monomials, you're looking for shared numerical and variable factors that can be combined to form a common factor.

Numerical common factors refer to the greatest common divisor among the coefficients of the given monomials. In the process of finding the GCF, identifying these numerical factors is a crucial first step. For example, consider the coefficients 10, 20, and 45 from our monomials. To discover their greatest common factor, you need to list the prime factors of each:

• 10 = 2 * 5
• 20 = 2^2 * 5
• 45 = 3 * 3 * 5

The common prime factor here is 5. Thus, the greatest common factor among the numerical coefficients is 5. Once you have identified this, you can move on to the variables and determine their lowest powers.

When seeking the GCF of monomials, identifying the lowest power of variables that appear in all monomials is essential. This ensures that the GCF captures what each term shares in its construction.In our exercise, look at the variables \(x\) and \(z\) that are present in all three monomials:

• For \(x\): The monomials have \(x^3\), \(x^2\), and \(x^1\). The lowest power is \(x^1\).
• For \(z\): The terms include \(z^3\), \(z^5\), and \(z^3\). The smallest power here is \(z^3\).

The variable \(y\) is not universal across all monomials, so it is excluded. By combining the lowest powers of \(x\) and \(z\), we construct part of the GCF. These lowest powers, combined with the numerical common factor, form the full GCF: \(5xz^3\). This reflects the shared components among the monomials.