Factor Completely:
1000p^3 - 1
(One thousand P cubed minus one)
That is the difference of two cubes: (10p)^3 and 1^3
One of the factors is (10p -1).
Divide that into 1000p^3 -1 for the other factor. Use polynomial long division. (The answer will start out 100 p^2 + 10p + ? ). Then see if you can factor that.
Thank You So Much!
To factor completely, we need to look for patterns or special identities that could help us rewrite the expression in a factored form. In this case, we can recognize the given expression as a difference of cubes.
The difference of cubes formula states that if we have an expression of the form a^3 - b^3, it can be factored as (a - b)(a^2 + ab + b^2).
In our given expression, we have 1000p^3 - 1. To apply the difference of cubes formula, we can rewrite 1000p^3 as (10p)^3 and 1 as 1^3:
(10p)^3 - 1^3
Now we can identify a = 10p and b = 1:
(a - b)(a^2 + ab + b^2)
Substituting a and b:
(10p - 1)((10p)^2 + (10p)(1) + 1^2)
Simplifying further:
(10p - 1)(100p^2 + 10p + 1)
Therefore, 1000p^3 - 1 can be factored completely as (10p - 1)(100p^2 + 10p + 1).