Question

Factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
21p10q2–21p7q3+7p7q2

Answer 1

To factor out the greatest common factor (GCF) from the polynomial \( 21p^{10}q^{2} - 21p^{7}q^{3} + 7p^{7}q^{2} \), we first need to determine the GCF of the coefficients and the variables.

1. **Coefficients**: The coefficients are 21, -21, and 7. The GCF of these numbers is 7.

2. **Variable \( p \)**: The powers of \( p \) are \( p^{10} \), \( p^{7} \), and \( p^{7} \). The GCF here is the lowest power, which is \( p^{7} \).

3. **Variable \( q \)**: The powers of \( q \) are \( q^{2} \), \( q^{3} \), and \( q^{2} \). The GCF here is also \( q^{2} \).

Now, we can combine these findings: the GCF of the entire polynomial is \( 7p^{7}q^{2} \).

Now we factor this GCF out of each term in the polynomial:

\[
21p^{10}q^{2} = 7p^{7}q^{2} \cdot 3p^{3}
\]

\[
-21p^{7}q^{3} = 7p^{7}q^{2} \cdot (-3q)
\]

\[
7p^{7}q^{2} = 7p^{7}q^{2} \cdot 1
\]

Putting this all together, we have:

\[
21p^{10}q^{2} - 21p^{7}q^{3} + 7p^{7}q^{2} = 7p^{7}q^{2} \left( 3p^{3} - 3q + 1 \right)
\]

So, the factored form of the expression is:

\[
7p^{7}q^{2} (3p^{3} - 3q + 1)
\]