how do i factor:
(p+q)^3 - (p-q)^3
This is the difference of two squares of the standard form
a^2+b^2=(a+b)(a-b)
where a= p+q
b=p-q
I don't get it
To factor the expression (p+q)^3 - (p-q)^3, we can use the identity for the difference of cubes. The identity states that for any two numbers a and b, (a^3 - b^3) can be factored as (a - b)(a^2 + ab + b^2).
Let's apply this identity to our expression:
(p+q)^3 - (p-q)^3 = [(p+q) - (p-q)][(p+q)^2 + (p+q)(p-q) + (p-q)^2]
Simplifying further, we have:
(p+q)^3 - (p-q)^3 = [2q][(p+q)^2 + (p^2 - q^2) + (p-q)^2]
Expanding the squared terms:
= 2q[p^2 + 2pq + q^2 + p^2 - q^2 + p^2 - 2pq + q^2]
Combining like terms:
= 2q[3p^2 + 2q^2]
Finally, we have factored the expression as:
(p+q)^3 - (p-q)^3 = 2q(3p^2 + 2q^2)