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Factor out the greatest common factor and simplify the factors, if possible.

[tex]3x^8 - 9x^7 + 18x^6[/tex]

Select the correct choice below and fill in any answer boxes in your choice:

A. [tex]3x^8 - 9x^7 + 18x^6 =[/tex] [tex]\square[/tex]
(Type your answer in factored form. Simplify your answer.)

B. The expression is not factorable.

Answer :

We start with the expression:

[tex]$$
3x^8 - 9x^7 + 18x^6.
$$[/tex]

Step 1. Identify the Greatest Common Factor (GCF):

- Coefficients: The coefficients are 3, -9, and 18. The GCF of these numbers is 3.
- Variable Part: The terms have exponents 8, 7, and 6 on [tex]$x$[/tex]. The smallest exponent is 6, so we have a common factor of [tex]$x^6$[/tex].

Thus, the GCF for the entire expression is [tex]$3x^6$[/tex].

Step 2. Factor Out the GCF:

Factor [tex]$3x^6$[/tex] out of each term:

- For [tex]$3x^8$[/tex]:
[tex]$$
3x^8 = 3x^6 \cdot x^2.
$$[/tex]

- For [tex]$-9x^7$[/tex]:
[tex]$$
-9x^7 = 3x^6 \cdot (-3x).
$$[/tex]

- For [tex]$18x^6$[/tex]:
[tex]$$
18x^6 = 3x^6 \cdot 6.
$$[/tex]

Step 3. Write the Expression in Factored Form:

After factoring out the GCF, the expression becomes:

[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]

Therefore, the simplified factored form of the expression is:

[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]

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