To solve the expression \( 27w^8 + 9w^4 - 18w^2 \), we will first identify the greatest common factor (GCF) of the terms, and then factor the expression.
Step 1: Find the GCF
Let's look at the coefficients of each term:
- The coefficients are 27, 9, and -18.
- The GCF of 27, 9, and 18 is 9.
Next, we look at the variable part:
- The terms are \( w^8 \), \( w^4 \), and \( w^2 \).
- The GCF of \( w^8, w^4, \) and \( w^2 \) is \( w^2 \).
Thus, the GCF of the entire expression \( 27w^8 + 9w^4 - 18w^2 \) is \( 9w^2 \).
Step 2: Factor out the GCF
Now that we have the GCF, we can express the original expression in its factored form:
\[ 27w^8 + 9w^4 - 18w^2 = 9w^2(3w^6 + w^2 - 2) \]
Now, let's summarize the results:
GCF: \( 9w^2 \)
Factored Form: \( 9w^2(3w^6 + w^2 - 2) \)
So, the answers are:
GCF: \( 9w^2 \)
Factored Form: \( 9w^2(3w^6 + w^2 - 2) \)