The quadratic polynomial x^4 + 5x^3 + 8x^2 + 8x -8 can be factored into two prime quadratics of the form (x^2 + ax + b)(x^2 + cx + d). Find (a,b,c,d) if a, b, c, and d are intergers.
Hello, Mr. Green,
honestly: by trial and error, because to do this system of non-linear equations looks like a life sentence.
1. You got: all a,b,c,d should be integers. Now take the last equation and solve for b = -8/d. That means d can only be in {-8, -4, -2, -1, 1, 2, 4, 8}
2. Choose a value for d; then you have b, then you can calculate c with the 3rd equation and last a.
You'll get pretty soon the only possible solution - but I didn't "solve" this system of equations.
EB
for confirmation only: [a, b, c, d] = [2, 4, 3, -2]
This is a pretty mean problem in general. I do have a way to cut down on some of the work, though.
I presume if you are doing this you are familiar with modular mathematics? (This seems to be my favorite trick recently.)
First note that b = -8, -4, -2, -1, 1, 2, 4, or 8.
Now write this polynomial (mod 3). (I tried it first with mod 2, but didn't get quadratic factors.)
(mod 3)
= (mod 3)
= (mod 3)
= (mod 3)
Comparing the first factor with we get b = 1 (mod 3).
Thus b = -8, -2, 1, or 4. Because d = 1, 4, -8, or -2 respectively we can arbitrarily decide that b = 1 or 4.
The rest can be done by applying these two possibilities to the problem to see what happens and significantly cuts down the work in finding a and c.
It's not a great solution, but could cut down the time if you have another problem like this.
-Dan