Factor out:
1-125b^3
5b -1 is one of the factors. (Consider that an informed guess). Divide that into 1-125b^3 for the other (binomial) factor.
To factor out 1-125b^3, we can make use of the binomial factor theorem.
First, let's consider the informed guess you provided: 5b - 1 is one of the factors. This means that if we multiply this factor by another binomial factor, it should give us 1-125b^3.
Next, divide 1-125b^3 by 5b - 1. We can use long division to do this:
_________
5b - 1 | 1 - 125b^3
To start, divide 1 by 5b:
_________
5b - 1 | 1 - 125b^3
- (1 - 125b^3)
___________
Next, multiply the divisor, 5b - 1, by -125b^2 and subtract it from our previous step:
_________
5b - 1 | 1 - 125b^3
- (1 - 125b^3)
___________
0 - 125b^2
Now, divide -125b^2 by 5b:
_________
5b - 1 | 1 - 125b^3
- (1 - 125b^3)
___________
0 - 125b^2
- (0 - 125b^2)
__________________
0
Since we obtain a remainder of 0, it means that our informed guess, 5b - 1, is indeed a factor of 1-125b^3.
Thus, we factor out 1-125b^3 as:
1 - 125b^3 = (5b - 1)(-125b^2)
Therefore, (5b - 1) is one of the factors, and the other factor is (-125b^2).