Factor out the greatest common factor from the following polynomial:

[tex]\[7x^7 - 35x^6 + 56x^5\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]$7x^7 - 35x^6 + 56x^5 =$[/tex] [tex]$\square$[/tex] (Type your answer in factored form.)

B. The polynomial has no common factor other than 1.

To factor out the greatest common factor (GCF) from the polynomial [tex]$7x^7 - 35x^6 + 56x^5$[/tex], let's follow these steps:

1. Identify the coefficients:
- The coefficients in the polynomial are: 7, -35, and 56.

2. Find the greatest common factor of the coefficients:
- The coefficient factors are:
- 7: [tex]$7$[/tex]
- -35: [tex]$7 \times -5$[/tex]
- 56: [tex]$7 \times 8$[/tex]
- The greatest common factor of 7, -35, and 56 is 7.

3. Observe the variable terms:
- The variable terms are: [tex]$x^7$[/tex], [tex]$x^6$[/tex], and [tex]$x^5$[/tex].
- The greatest power of [tex]$x$[/tex] that is common to all these terms is [tex]$x^5$[/tex].

4. Determine the GCF of the entire expression:
- Combining the factors found, the greatest common factor of the polynomial is [tex]$7x^5$[/tex].

5. Factor out the GCF from each term of the polynomial:
- [tex]$7x^7 \div 7x^5 = x^2$[/tex]
- [tex]$-35x^6 \div 7x^5 = -5x$[/tex]
- [tex]$56x^5 \div 7x^5 = 8$[/tex]

6. Write the polynomial in factored form:
- After factoring out the GCF, the polynomial becomes:
[tex]\[
7x^5(x^2 - 5x + 8)
\][/tex]

Therefore, the correct factored form of the polynomial [tex]$7x^7 - 35x^6 + 56x^5$[/tex] is:

[tex]\[
7x^5(x^2 - 5x + 8)
\][/tex]

This matches option A: [tex]$7x^7 - 35x^6 + 56x^5 = \boxed{7x^5(x^2 - 5x + 8)}$[/tex].