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The greatest common factor (GCF) is the largest number that evenly divides into each of the terms of the polynomial. For polynomials, you look for the highest number that divides the coefficients and the highest power of any variable common to all terms.

In our example with the polynomial expression, we have: \(12x^3-10x\).
The coefficients 12 and -10 both have a GCF of 2, as 2 is the highest number that divides both 12 and 10 evenly.
For the variable part, we look at \(x^3\) and \(x\). The highest power of \(x\) common in both terms is just \(x\).

So the GCF for the entire polynomial is \(2x\).

Identifying the GCF is a crucial first step that simplifies factoring the polynomial.

Once you've identified the GCF, the next step involves rewriting the polynomial using this factor. This breaks down the original terms into more manageable parts.

Here’s how it looks for our polynomial \(12x^3 - 10x\):

• First term: \(12x^3\) can be rewritten as \(2x \times 6x^2\), because multiplying \(2x\) with \(6x^2\) returns the original term \(12x^3\).
• Second term: \(-10x\) becomes \(2x \times -5\), because multiplying \(2x\) with \(-5\) brings back the term \(-10x\).

Once each term is rewritten this way, the polynomial can then be regrouped to show the factorization clearly, like this: \(2x(6x^2 - 5)\).

These factoring techniques help in breaking down complex polynomial expressions into simpler pieces.

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, in the given polynomial \(12x^3 - 10x\),

• \(12x^3\): This means 12 multiplies the cube of the variable \(x\).
• \(-10x\): This term represents -10 multiplied by the variable \(x\).

To factor polynomials effectively, one must understand how these terms fit together. Restructuring them using the GCF can significantly simplify the equation.

Once you rewrite the terms with the GCF, you see how they simplistically fit under a common factor. This makes it easier to work with and understand polynomials.

For our example, the final factored form \(2x(6x^2 - 5)\) shows the polynomial’s components in a compact and understandable manner. Here, \(2x\) is the common factor across the terms, maintaining the integrity of the polynomial's overall structure.