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.
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Coefficients: The coefficients are 21, -21, and 7. The GCF of these numbers is 7.
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Variable \( p \): The powers of \( p \) are \( p^{10} \), \( p^{7} \), and \( p^{7} \). The GCF here is the lowest power, which is \( p^{7} \).
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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:
\[ 7p^{7}q^{2} (3p^{3} - 3q + 1) \]
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