Write f(x)=x^4-12x^3+59x^22-138x+130 as a product of linear factors.

I listed the factors of 130 and used synthetic division, but none of the remainders came out to be 0. I graphed the function, but the function doesn't pass through the x-axis. I figured I could find the zeros and work out the problem that way, but obviously it didn't work. Does the function have imagenary numbers when factored? I'm stumped on this problem, so if any of you could help me by showing the answer and by providing the steps the took to obtain that answer, I'd really appreciate it! Thanks!

To factor the given polynomial, we can start by looking for any rational roots using the rational root theorem. According to the rational root theorem, any rational root of the polynomial f(x) must be in the form of p/q, where p is a factor of the constant term (130 in this case) and q is a factor of the leading coefficient (1 in this case).

You mentioned that you listed the factors of 130, but none of the remainders came out to be 0 after using synthetic division. This suggests that the polynomial may not have any rational roots.

If the polynomial does not have any rational roots, it means that it cannot be factored into linear factors with real coefficients. Therefore, we cannot express it as a product of linear factors.

However, we can still attempt to find the complex roots of the polynomial. This can be done by using numerical methods such as the Newton-Raphson method or by using graphing calculators or software to find the x-intercepts (zeros) on the graph. These methods can provide an approximation of the complex roots.

In summary, since the polynomial does not have any rational roots, it can't be factored into linear factors with real coefficients. The complex roots of the polynomial can be found using numerical methods or by graphing the function to approximate the zeros.