MHB What is the Factorization of p^3 - q^3 -p(p^2 - q^2) + q(p - q)^2?

[mathdad]

Factor.

p^3 - q^3 -p(p^2 - q^2) + q(p - q)^2

Solution:

p^3 - q^3 - p^3 + pq^2 + q(p^2 -2pq + q^2)

p^3 - q^3 - p^3 + pq^2 + qp^2 -2pq^2 + q^3

pq^2 + qp^2 - 2pq^2

-pq^2 + qp^2

qp^2 - pq^2

pq(p - q)

Correct?

 

[MarkFL]

Let's see:

$$p^3-q^3-p(p^2-q^2)+q(p-q)^2=(p-q)\left(\left(p^2+pq+q^2\right)-p(p+q)+q(p-q)\right)=(p-q)(p^2+pq+q^2-p^2-pq+pq-q^2)=pq(p-q)\quad\checkmark$$

 

[mathdad]

It feels good to find the right answer.