We are tasked with factoring the expression $125 + b^3$.

We recognize that $125$ is a perfect cube because: $125 = 5^3$ So, the expression $125 + b^3$ becomes: $5^3 + b^3$

This matches the form of a sum of cubes, which can be factored using the identity: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ In this case, $a = 5$ and $b = b$, so applying the formula:

$125 + b^3 = (5 + b)(25 - 5b + b^2)$

Thus, the fully factored form of $125 + b^3$ is: $(5 + b)(25 - 5b + b^2)$

Would you like any further explanation or details on this?

Related Questions:

1. How would you factor a difference of cubes like $b^3 - 125$?
2. Can this factorization be applied to any sum of cubes?
3. What happens if $b$ is negative in this sum of cubes?
4. How do we derive the identity for sum and difference of cubes?
5. Are there similar factorization identities for higher powers?

Tip:

Always look for perfect squares or cubes in expressions—they often hint at special factorizations.