Question

Factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
4p7+50p4q5+16p4q4+6q9

Answer 1

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.

1. **Identify the coefficients**: The coefficients are \(4\), \(50\), \(16\), and \(6\). The GCF of these numbers is \(2\).

2. **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:

\[
2p^4(2p^3 + 25q^5 + 8q^4 + 3q^9)
\]