Factorize the following expression

[abrowaqas]

Homework Statement

Expression is
p^3-(x^2+xy+y^2)p+xy^2+x^2y=0 where ^ shows power

Homework Equations

The Attempt at a Solution

the solution is
(p-x)(p-y)(p+x+y)=0

 

[Char. Limit]

So, what's the question? I see an expression, told to factorize, and I see the solution. What's your question?

 

[Mark44]

The OP would like to factor the expression on the left side of the equation, and is showing the solution shown in the book, but doesn't know how to get the factorization that is shown.

I believe the best way to crack this one is to use a known formula for factoring a cubic. Here's a discussion of several techniques - http://en.wikipedia.org/wiki/Cubic_equation. See the section titled "Cardano's method" on the linked-to page.

Factoring cubic equations is a lot more complicated than factoring quadratics, but the saving grace here is that there is no quadratic (p²) term in your equation, which should make the work a little easier.

 

[vela]

Try adding and subtracting pxy to what's inside the parentheses to get

$$\begin{align*}<br /> 0 &= p^3-(x^2+xy+y^2+xy-xy)p+xy^2+x^2y \\<br /> &=p^3-(x^2+2xy+y^2)p + pxy + xy^2+x^2y \\<br /> &=p^3-(x+y)^2p+pxy+xy^2+x^2y<br /> \end{align*}$$

Then pull common factors out of the first two terms and the last two terms. That should get you started.