Polynomial Factorization for Integers: Finding Real and Complex Roots

[Mentallic]

Homework Statement

In the polynomial [tex]x^5+22x^3-34x^2+117x-306[/tex] given that the roots on the real and imaginary plane are all integers, factorize the polynomial into real linear and quadratic factors.

The Attempt at a Solution

I was able to find the real factor, which is x=2 and then I reduced the polynomial to this:

[tex](x-2)(x^4+2x^3+26x^2+18x+153)[/tex]

But this quartic has all complex roots, and I'm unsure how to find them given the information above. Please help!

 

[Mentallic]

Oh and I forgot to mention that since the polynomial has real coefficients, the complex roots will have complex conjugate roots also, so this can be expressed as follows:

[tex]x^4+2x^3+26x^2+18x+153 \equiv (x-(a+ib))(x-(a-ib))(x-(c+id))(x-(c-id))[/tex]

where, a,b,c,d all integers (from the information in the question).

Expanding these complex linear factors into real quadratic factors gives:

[tex](x^2-2a+a^2+b^2)(x^2-2c+c^2+d^2)[/tex]

I could try expand this and equate the coefficients with the original quartic, but this is quite a lot of effort considering the question should have been answered relatively quickly.

 

[Mentallic]

bump.

 

[Bohrok]

The last term in each of the quadratic factors have to be factors of 153, so either 3 and 51 or 9 and 17, but the last pair is the obvious choice. Then you could write out the quadratic factors as (x² + ax + 9)(x² + bx + 17)
Now you don't have to multiply out the whole thing but only part of it. Multiplying it out, you would get two terms with x³, three with x², and two with x. Multiply the terms that give you the x³ and x terms and equate them with the coefficients in your quartic. You'll easily get a and b in the two quadratic factors above and you're done.

 

[Mentallic]

Oh I was just told today that it is acceptable to use Maple to solve this problem, considering no one else could answer it within the 4 minute allocated time limit.

Anyway, not just with guessing but from the quadratics I expressed, we also have:

[tex](a^2+b^2)(c^2+d^2)=153[/tex] where [tex]a,b,c,d E Z[/tex]

then looking at the possible factors -

1 and 153
3 and 51
9 and 17

It can be concluded that the only possible pair that can be expressed as above is 9 and 17, with [tex]a=0, b=\pm 3[/tex] (since it has to be an imaginary root) and [tex]c,d=\pm 4,\pm 1[/tex].

From there, using the other relationships, the results can be found.