How do you write # f(x)=x^5-3x^4-4x^3+28x^2-37x+15# in factored form?
Use numerical methods as shown below.
Since there is no direct formula (that i am aware of) to find the roots of a 5th degree polynomial, I would suggest using numerical methods to solve this problem, of which there are many.
A good simple formula in numerical analysis which converges rather quickly, is Newton's Method of root finding.
It involves selecting an arbitrary starting value
The algorithm to use is :
Then select another arbitrary value (a distance away from the first root found) and repeat Newton's algorithm and converge to a 2nd root.
Continuing in this way, you will eventually find all 5 roots (if they exist) and would have the full factors of the quintic polynomial.
I have performed the iterations with initial value 0 and after 2 iterations found convergence to root value approximately 0,657 and hence (x-0,657) is a factor.
You may continue in this fashion to find the other factors.
Please ask me if you still need further assistance.
Trevor is correct that quintic polynomials are not solvable by simple formulae - though there are some spectacularly complex methods to find solutions in terms of a special kind of radical called a Bring radical. However, this is not a general quintic and this particular one is solvable.
By the rational root theorem, any rational roots of
That means that the only possible rational roots of
Let's try some:
The remaining cubic factor is also zero when
So there's another factor of
The discriminant of
So this quadratic factor has no simpler factors with Real coefficients.
We can write: