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------------------------------------------------ Find the greatest common factor of these three expressions:

[tex]28x, 70x^5, \text{ and } 14x^4.[/tex]

To find the greatest common factor (GCF) of the expressions [tex]$28x$[/tex], [tex]$70x^5$[/tex], and [tex]$14x^4$[/tex], we need to look at both the numerical coefficients and the variable parts separately.

1. Numerical Coefficients:
- The coefficients of the expressions are 28, 70, and 14.
- To find the GCF of these numbers, we find the largest number that divides each of them without leaving a remainder.
- The factors of 28 are: 1, 2, 4, 7, 14, 28.
- The factors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70.
- The factors of 14 are: 1, 2, 7, 14.
- The common factors of 28, 70, and 14 are: 1, 2, 7, 14.
- The greatest of these common factors is 14.

2. Variable Parts:
- The variable parts are [tex]$x$[/tex], [tex]$x^5$[/tex], and [tex]$x^4$[/tex].
- We look for the lowest power of [tex]$x$[/tex] that appears in all expressions, which is [tex]$x^1$[/tex] (or just [tex]$x$[/tex]).

3. Combine the Results:
- The GCF of the coefficients is 14.
- The GCF of the variable parts is [tex]$x$[/tex].
- Therefore, the greatest common factor of the entire expressions is [tex]$14x$[/tex].

So, the greatest common factor of the expressions [tex]$28x$[/tex], [tex]$70x^5$[/tex], and [tex]$14x^4$[/tex] is [tex]$14x$[/tex].