This is a "with the help of your classmates" problem. I am doing self study, so I am going to ask for a little help. According to a certain theorem, if a polynomial like x^4+4 has no real zeros, its factors are irreducible quadratics of the form (x^2 + ax + b)(x^2 + cx + d). We are then asked to apply essentially the same strategy we used with partial fraction decomposition to arrive at a system of equations which we can use to solve for a,b,c and d. That strategy is to take our largely symbolic quadratics, multiply them out, combine like terms, and the equate the coefficient terms (a,b,c and d as they appear naturally in the product's like terms) of the product with the coefficients of the original term that the quadratics are said to be factors of, namely, x^4 + 4. These equations then become our system of equations and should allow us to solve for a,b,c and d.
Fine. If you multiply the two largely symbolic quadratics out and combine like terms, you get
x^4 + cx^3 + dx^2 + ax^3 + acx^2 + adx + bx^2 + bcx + bd
combining like terms, you express them as
1 = 1 ?? (with fraction decomposition, you usually have a higher degree in the numerator so a higher degree in your symbolic terms, but not so here. This means there's no symbol for the coefficient of x^4.)<edit: this is a wrong or nonsensical comment--the coefficients of the symbolic expansion in fraction decomposition problems are equated to the coefficients of the *numerator* of the original term, which is often of a lower degree for some reason. The issue here is that x^4 provides no help for some reason>
a + c = 0 (there is no x^3 term in the original term, so its coefficient is zero)
a + b + c + d= 0 (acx^2 and bx^2 and dx^2 are the same as (a + b + c + d) * x^2 !?!)
a + b + c + d = 0 (this and the one above are the same so only one need be included I know)
bd = 4
And that would be the system, but it is wrong because I looked at the answer (x^2 - 2x + 2)(x^2 +2x + 2). This means that a = -2, and b,c and d = 2. That doesn't hold up in the system above. Also, I don't know what to do with db = 4 in a matrix. That is if my matrix is now
1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 ?? ?? ?? ?? 4
how do I make sense of the last entry. <edited the last entry--bd = 4 is the same as the rest, b is a coefficient of 1, as is d, so (b + d) * 1 = 4 is the same as bd = 4, i hope> ack this isn't right. Ok so I still don't understand, how do I use bd = 4 in the matrix.