Advanced book on Mathematics Olympiad

2.2 Polynomials 47

147.LetP(x)be a polynomial of odd degree with real coefficients. Show that the
equationP(P(x))=0 has at least as many real roots as the equationP(x)=0,
counted without multiplicities.

148.Determine all polynomialsP(x)with real coefficients for which there exists a
positive integernsuch that for allx,

P

(

x+

1

n

)

+P

(

x−

1

n

)

= 2 P(x).

149.Find a polynomial with integer coefficients that has the zero

√

2 +^3

√

3.

150.LetP(x)=x^4 +ax^3 +bx^2 +cx+dandQ(x)=x^2 +px+qbe two polynomials
with real coefficients. Suppose that there exists an interval(r, s)of length greater
than 2 such that bothP(x)andQ(x)are negative forx∈(r, s)and both are positive
forxs. Show that there is a real numberx 0 such thatP(x 0 ) < Q(x 0 ).

151.LetP(x)be a polynomial of degreen. Knowing that

P(k)=

k

k+ 1

,k= 0 , 1 ,...,n,

findP (m)form>n.

152.Consider the polynomials with complex coefficients

P(x)=xn+a 1 xn−^1 +···+an

with zerosx 1 ,x 2 ,...,xnand

Q(x)=xn+b 1 xn−^1 +···+bn

with zerosx 12 ,x^22 ,...,x^2 n. Prove that ifa 1 +a 3 +a 5 +···anda 2 +a 4 +a 6 +···

are both real numbers, then so isb 1 +b 2 +···+bn.

153.LetP(x)be a polynomial with complex coefficients. Prove thatP(x)is an even
function if and only if there exists a polynomialQ(x)with complex coefficients
satisfying

P(x)=Q(x)Q(−x).

2.2.2 Viète’s Relations.........................................

From the Gauss–d’Alembert fundamental theorem of algebra it follows that a polynomial

P(x)=anxn+an− 1 xn−^1 +···+a 0