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The Greatest Common Factor (GCF) of a set of numbers is the largest number that divides all the numbers in the set without leaving a remainder. For example, consider the terms in our trinomial: \(25z^4\), \(5z^3\), and \(z^2\). We look at the coefficients (25, 5, and 1) and the variable part (\(z^4\), \(z^3\), and \(z^2\)).

The GCF of the coefficients 25, 5, and 1 is 1 since 1 is the only number that can divide all three.

Next, we identify the smallest power of the variable common to all terms. The powers of \(z\) in our terms are 4, 3, and 2 respectively. The smallest power of \(z\) that is common to all terms is \(z^2\).

Therefore, the GCF of the trinomial \(25z^4 + 5z^3 + z^2\) is \(z^2\). Factoring the GCF from the expression simplifies the problem and makes further factorization easier.

A quadratic polynomial is a polynomial of degree 2, typically written in the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants.

In our case, after factoring out the GCF \(z^2\) from \(25z^4 + 5z^3 + z^2\), we are left with the quadratic polynomial \(25z^2 + 5z + 1\). This polynomial has a leading coefficient of 25, a middle term coefficient of 5, and a constant term of 1.

Not all quadratic polynomials can be easily factored further. The standard method to check if a quadratic polynomial can be factored is to apply the quadratic formula or complete the square method. However, in this instance, the polynomial \(25z^2 + 5z + 1\) is already in its simplest form and cannot be factored any further using real numbers.

Trinomial factorization involves expressing a trinomial as a product of two binomials or identifying a common factor among the terms of the trinomial.

For the given trinomial \(25z^4 + 5z^3 + z^2\), the first step is to find and factor out the greatest common factor (GCF), which we identified as \(z^2\).

When we factor out \(z^2\), we get \(z^2(25z^2 + 5z + 1)\).

Next, we look at the quadratic polynomial inside the parentheses: \(25z^2 + 5z + 1\). This step is crucial because sometimes the quadratic polynomial can be factored further.
In our case, it cannot be factored further using real numbers.
Thus, the completely factored form of the original trinomial is \(z^2(25z^2 + 5z + 1)\).
Always remember to check if the quadratic polynomial can be simplified.