Factor each expression
(X+5)^3-(2x+1)^3
To factor the given expression (X + 5)^3 - (2x + 1)^3, we can use the difference of cubes formula, which is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a = (X + 5) and b = (2x + 1). So, the expression can be factorized as:
((X + 5) - (2x + 1))((X + 5)^2 + (X + 5)(2x + 1) + (2x + 1)^2)
Now, let's simplify this expression further:
((X + 5) - (2x + 1))((X + 5)^2 + (2x + 1)(X + 5) + (2x + 1)^2)
Simplifying the first part:
(X + 5) - (2x + 1) = X + 5 - 2x - 1 = -x + 4
Now, let's simplify the second part:
((X + 5)^2 + (2x + 1)(X + 5) + (2x + 1)^2)
= (X^2 + 10X + 25) + (2x + 1)(X + 5) + (4x^2 + 4x + 1)
= X^2 + 10X + 25 + (2x^2 + 12x + 5x + 25) + 4x^2 + 4x + 1
= X^2 + 2x^2 + 10X + 12x + 5x + 25 + 4x^2 + 4x + 1
= 7x^2 + 21x + 26
Combining both parts:
(-x + 4)(7x^2 + 21x + 26)
Therefore, the factored form of the expression (X + 5)^3 - (2x + 1)^3 is (-x + 4)(7x^2 + 21x + 26).