Can x3 be factored into common quadratic factor for P(x) and Q(x)?

[songoku]

Homework Statement

Find the value of m and n, where m and n are integer, so that P(x) = x³ + mx² – nx - 3m and
Q(x) = x³ + (m – 2) x² –nx – 3n have common quadratic factor.

Homework Equations

The Attempt at a Solution

Is m = n = 0 one of the solution?

If m = n = 0,then :
P(x) = x³ = x² (x)

Q(x) = x³ - 2x² = x² (x-2)

Can I say that they have common quadratic factor, which is x² ?

The point is : Am I right to say x³ can be be factorized to x² x or even (x) (x) (x) ?

Thanks

 

[Mark44]

That works, but there might be another solution with both functions having a quadratic factor of x² + bx + c. I've filled up a couple of pieces of paper without finding it, though.

 

[vela]

m=5 and n=3 works.

 

[vela]

If p(x) and q(x) have a common quadratic factor, they can be written

$$p(x) = (x+a)(x^2+cx+d)$$
$$q(x) = (x+b)(x^2+cx+d)$$

so that

$$p(x)-q(x) = (a-b)(x^2+cx+d)$$.

Try plugging the given polynomials into the LHS and match coefficients to determine a, b, c, d, m, and n.

 

[songoku]

Hi Mark and vela

Thanksssss !