Advanced book on Mathematics Olympiad

46 2 Algebra

real, or complex coefficients are denoted, respectively, byZ[x],Q[x],R[x], andC[x].
A numberrsuch thatP(r)= 0 is called a zero ofP(x), or a root of the equation
P(x)=0. By the Gauss–d’Alembert theorem, also called the fundamental theorem
of algebra, every nonconstant polynomial with complex coefficients has at least one
complex zero. Consequently, the number of zeros of a polynomial equals the degree,
multiplicities counted. For a numberα,P(α)=anαn+an− 1 αn−^1 +···+a 0 is called
the value of the polynomial atα.
We begin the section on polynomials with an old problem from the 1943 competition
of theMathematics Gazette, Bucharest, proposed by Gh. Buicliu.

Example.Verify the equality

3

√

20 + 14

√

2 +

3

√

20 − 14

√

2 = 4.

Solution.Apparently, this problem has nothing to do with polynomials. But let us denote

the complicated irrational expression byxand analyze its properties. Because of the cube

roots, it becomes natural to raisexto the third power:

x^3 = 20 + 14

√

2 + 20 − 14

√

2

+ 3

3

√

( 20 + 14

√

2 )( 20 − 14

√

2 )

(

3

√

20 + 14

√

2 +

3

√

20 − 14

√

2

)

= 40 + 3 x^3

√

400 − 392 = 40 + 6 x.

And now we see thatxsatisfies the polynomial equation

x^3 − 6 x− 40 = 0.

We have already been told that 4 is a root of this equation. The other two roots are

complex, and hencexcan only equal 4, the desired answer. 

Of course, one can also recognize the quantities under the cube roots to be the cubes

of 2+

√

2 and 2−

√

2, but that is just a lucky strike.

145.Given the polynomialP(x, y, z)prove that the polynomial

Q(x,y,z)=P(x, y, z)+P(y, z, x)+P(z, x, y)

−P(x, z, y)−P(y, x, z)−P(z, y, x)

is divisible by(x−y)(y−z)(z−x).

146.Find all polynomials satisfying the functional equation

(x+ 1 )P (x)=(x− 10 )P (x+ 1 ).