# Homework Help: Common quadratic factor

1. Jan 27, 2010

### songoku

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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

2. Jan 28, 2010

### Staff: Mentor

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.

3. Jan 28, 2010

### vela

Staff Emeritus
m=5 and n=3 works.

4. Jan 28, 2010

### vela

Staff Emeritus
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.

5. Jan 28, 2010

### songoku

Hi Mark and vela

Thanksssss !!