To factor \( x^{63} + 125y^3 \), we can recognize it as a sum of cubes. The expression can be rewritten as:
\[ x^{63} + (5y)^3 \]
We can apply the sum of cubes formula, which states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
In this case, let \( a = x^{21} \) (since \( (x^{21})^3 = x^{63} \)) and \( b = 5y \) (since \( (5y)^3 = 125y^3 \)). Using the sum of cubes formula:
- \( a + b = x^{21} + 5y \)
- \( a^2 = (x^{21})^2 = x^{42} \)
- \( -ab = -x^{21}(5y) = -5x^{21}y \)
- \( b^2 = (5y)^2 = 25y^2 \)
Now, we can combine these to create the factored form:
\[ x^{63} + 125y^3 = (x^{21} + 5y)(x^{42} - 5x^{21}y + 25y^2) \]
Thus, the factored form of \( x^{63} + 125y^3 \) is:
\[ (x^{21} + 5y)(x^{42} - 5x^{21}y + 25y^2) \]