Question

Find all the polynomials P(X) such that P(P(X)) = P(X) + 7.

Answer 1

where - Solution: - det Plocy is Polynomial of degren, nis non negative Integer. P(P(x)) is also a Polynomial of degree Thon لیاہ (۱N P(P(x)) P(x) +7 both side of Polynomial Comparing degree =) degree of P(P6)) = degree of P(x) of n=1 -> g=n no 7 Poco is Polynomial of degree o. Tron if Pixs is Polynomial. constant des Plx)=4 taflR = 4+7 poco +7 and Then P(P(x)) = P(a)=x, a $2+7 Thus PlPix & Poxo +7 of Poo can not be Polynomial It Imply degree

Now del Pixo is Polynomial of degree I Trien Pixo = AX+B A to, and AB are Constants. Then P(P6) • Axt + AB+B - ALAY+B) +B Also P(x) +7 AX+B+7 Now given P (PO) = Poxox7 => ASC + ABIB AX+B+7 Comparing the coefficient of a cant constant Å=A, AB+ B = B+7 tem we get Solving these A=!, B=7 Polynomial - Poco +7 ane So there is only P(P(x) ) such that whese Plx) x+7 Answer