To factor out the greatest common factor (GCF) from the polynomial \(36f^9 + 36f^5 - 12f^2\), we first need to determine the GCF of the coefficients and the variable parts.
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Coefficients: The coefficients are 36, 36, and -12.
- The GCF of 36 and -12 is 12.
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Variable part: The terms have \(f^9\), \(f^5\), and \(f^2\).
- The GCF for the variable parts is \(f^2\) (the lowest power of \(f\)).
Combining these, the overall GCF of the polynomial is \(12f^2\).
Now, we can factor out the GCF:
\[ 36f^9 + 36f^5 - 12f^2 = 12f^2(3f^7 + 3f^3 - 1). \]
So, the factored form of the polynomial is:
\[ 12f^2(3f^7 + 3f^3 - 1). \]