I came across this question while studying the theory of equations. I also have the answer plus explanation given. But I could not understand them. Please Help.
Let $(a_1, a_2, a_3, a_4, a_5)$ denote a rearrangement of $(3, -5, 7, 4, -9)$, then the equation $$a_1\cdot x^4 + a_2\cdot x^3 + a_3\cdot x^2 + a_4\cdot x + a_5 = 0$$ has,
a) at least two real roots
b) all four real roots
c) only imaginary roots
d) none of these.
explanation. $x=1$ is always a root of the equation.
As $a_1, a_2,...,a_5$ are rearrangement of $3,-5,...,-9$ is it necessary to check whether all the possible equations have atleast two real roots or what? A short cut or concept behind the solution? Please help.