factor the polynomial completely.
125s to the third power + 1
This is the same as (5y)^3+1.
This is the type of equation that is called sum of cubes. There is a special formula for those, see your text.
To factor the polynomial completely, we can use the formula for the sum of cubes. The formula is:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In this case, we have 125s^3 + 1. Let's rewrite it as (5s)^3 + 1.
Using the formula, we can see that a = 5s and b = 1. Plugging these values into the formula, we get:
(5s)^3 + 1 = (5s + 1)((5s)^2 - (5s)(1) + 1^2)
= (5s + 1)(25s^2 - 5s + 1)
Therefore, the polynomial 125s^3 + 1 can be factored completely as (5s + 1)(25s^2 - 5s + 1).