Abel's theorem in problems and solutions - School of Mathematics

The complex numbers 73
where all the are arbitrary complex numbers, and has
at least one complex root.
268. Prove Bézout’s theorem 10 : If is a root of the equation
then the polynomial is divisible
by the binomial without remainder.
269. Prove that the polynomial where
can be represented in the form:
REMARK. Suppose that the polynomial decomposes into factors:
The left member of the equation is equal to 0 if and only if at least one of
the factors inside brackets is equal to 0 (cf., 195, 197). Hence the roots
of equation are the numbers and them alone.
270. Let be a root of the equation
where the are real numbers. Prove that the number the conjugate
of is also a root of that equation.
271. Suppose that the equation with real coefficients
has a complex root being not a real number. Prove that the polynomial
is divisible by a polynomial of second degree with real coefficients.
272. Prove that every polynomial with real coefficients can be written
in the form of a product of polynomials of first and second degree with
real coefficients.
REMARK. From the result of Problem 272 it follows that the sole
irreducible polynomials (cf., §2.3) over the field of real numbers are the
10 É. Bézout (1730–1783), French mathematician.