Question 1175008: factor the following:
f(x)=x^4-8x^3+22x^2-24x+7
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website! .
Actually, this polynomial can not be factored, because IT HAS NO rational roots.
To prove it, use the Rational root theorem test.
For it, you should test if the divisors of the constant term "7" are roots of this polynomial.
In other words, you should check if some of the four integer numbers -1, 1, -7, 7 is the root of the polynomial.
It is easy to check - I did it using MS Excel spreadsheet in my computer.
The answer is "NO" : no one of these integer numbers is the root of the polynomial.
It means that the polynomial HAS NO rational roots,
and hence, CAN NOT be factorable over integer numbers.
Solved.
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What I said above, proves that the given polynomial has no linear factors over rational numbers.
But a potential opportunity still remains to factor it into the product of quadratic polynomials.
In the Internet, there are several sites offering free of charge solvers to factor polynomials.
I will not refer to concrete sites/solvers --- you can easily find them on your own IF YOU WISH.
I tried some of them (three or four) --- they all said that factoring is impossible over rational numbers.
One site was even more talkative: it even gave me the roots, that were irrational numbers.
It is the solver
https://www.wolframalpha.com/input/?i=x%5E4-8x%5E3%2B22x%5E2-24x%2B7
But its conclusion about potential factorability into the product of quadratic polynomial
with rational (or integer) coefficients was NEGATIVE, too : I M P O S S I B L E .
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