Divide 1-125b^3 by 5b - 1

You need to learn the technique of polynomial long division. It's just the same as if numbers were involved.

See http://www.sosmath.com/algebra/factor/fac01/fac01.html
for sevaral examples.
Hint: It will be a three-term polynomial. The first term will be
25 b^2 and the last term will be 1

To divide the polynomial 1 - 125b^3 by 5b - 1, we will use polynomial long division. Here are the steps to perform this division:

1. Write the dividend (1 - 125b^3) and divisor (5b - 1) in the long division format with the dividend on the top and the divisor on the left side.

________________________
5b - 1 | 1 - 125b^3

2. Start by dividing the first term of the dividend (1) by the first term of the divisor (5b), which gives us (1/5b) as the first term of the quotient. Write this term above the division symbol.

1/5b
________________________
5b - 1 | 1 - 125b^3

3. Multiply the divisor (5b - 1) by the quotient term (1/5b), and write the result below the dividend.

1/5b
________________________
5b - 1 | 1 - 125b^3
-(1 - b^3)
___________
-b^3 + 1

4. Subtract this result from the first term of the dividend. Bring down the next term (-125b^3) to the next line.

1/5b
________________________
5b - 1 | 1 - 125b^3
-(1 - b^3)
___________
-b^3 + 1
-( -b^3 + 1)

5. Repeat steps 2 to 4 until all terms have been divided. In this case, we will perform the division on the second term (-b^3).

1/5b
________________________
5b - 1 | 1 - 125b^3
-(1 - b^3)
___________
-b^3 + 1
-( -b^3 + 1)
____________
0

6. At this point, we have divided all the terms of the dividend. The quotient will be the terms written above the division symbol.

Quotient: 1/5b - 1/5

Hence, the division of 1 - 125b^3 by 5b - 1 is 1/5b - 1/5.