21) \( a^{3}+343 b^{3} \)

To factor the expression \( a^{3}+343 b^{3} \), recognize that \( 343 \) can be rewritten as \( 7^{3} \). Thus, the expression becomes \( a^{3}+(7b)^{3} \). This fits the sum of cubes formula \( x^{3}+y^{3} = (x+y)(x^{2}-xy+y^{2}) \), where \( x = a \) and \( y = 7b \). So, you can factor it as \( (a + 7b)(a^{2} - 7ab + (7b)^{2}) \), which simplifies further into \( (a + 7b)(a^{2} - 7ab + 49b^{2}) \). To see this in action, consider applying it to real-world scenarios like simplifying polynomial expressions in physics or engineering. Using factored forms can make calculations easier, especially when working with area, volume, or even forces. The more you practice factoring, the quicker you become at recognizing patterns and solving problems efficiently! Also, watch out for common mistakes like not recognizing cubes or forgetting to rewrite constants as cubes—always double-check your expressions!