To factor out the greatest common factor (GCF) from the polynomial \(4p^7 + 50p^4q^5 + 16p^4q^4 + 6q^9\), we first need to identify the GCF of all the terms.
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Identify the coefficients: The coefficients are \(4\), \(50\), \(16\), and \(6\). The GCF of these numbers is \(2\).
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Identify the variable factors:
- For \(p\): The term with the smallest power is \(p^4\) (from \(50p^4q^5\) and \(16p^4q^4\)).
- For \(q\): The term with the smallest power is \(q^0\) (meaning we don’t factor \(q\) out at all because \(q^9\) is the highest power and is not present in all terms).
So the overall GCF is \(2p^4\).
Now we can factor \(2p^4\) out of each term:
\[ 4p^7 = 2p^4 \cdot 2p^3 \] \[ 50p^4q^5 = 2p^4 \cdot 25q^5 \] \[ 16p^4q^4 = 2p^4 \cdot 8q^4 \] \[ 6q^9 = 2p^4 \cdot 3q^9/p^4 \quad \text{(since } p^4 \text{ is a factor of } 6q^9 \text{, we include } and consider the overall non-existence of } p^4 \text{ here)}. \]
Now writing the polynomial with \(2p^4\) factored out:
\[ 4p^7 + 50p^4q^5 + 16p^4q^4 + 6q^9 = 2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9/p^4) \]
However since \(6q^9\) does not contain \(p^4\), we can also simplify it differently (based on remaining powers):
\[ == 2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9) \]
Now, notice \(6q^9\) is a separate consideration. Re-arranging a bit gives us:
\[ = 2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9) \]
Thus, the factored form, without leaving \(p^4\) in the last term would give:
\[ 2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9) \]
Therefore, the answer is:
\[ 2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9) \]
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