A polynomial is said to be palindromic if its coefficients a

A polynomial is said to be palindromic if its coefficients are the same read left to right or right to left. Show that a general palindromic polynomial in P_3 is of the form a + bx + bx^2 + ax^3. Let the set of palindromic polynomials in P_3 be denoted S. By making a correspondence between elements of P_3 and four-tuples (a, b, c, d), write the subset of R^4 corresponding to S as a span. What is the dimension of this subspace? Show from first principles that the set S is closed under vector addition.

Solution

(a)A palindromic polynomial is a self reciprocal polynomial. Let a polynomial in P₃ be p(x) = a₀ +a₁x + a₂x² + a₃x³ . Its reciprocal polynomial is p^*(x) = a₃ + a₂x +a₁x² +a₀x³. If p(x) is a palindromic polynomial, then we have p(x)= p^*(x) i.e. a₀ +a₁x + a₂x² + a₃x³ = a₃ + a₂x +a₁x² +a₀x³ . Hence, a₀ = a₃ = a (say) and a₁ = a₂ = b (say). Then, the polynomial p(x) changes to a+ bx +bx² +ax³. Thus, a general pallindromic polynomial in P₃ is of the form a+ bx +bx² +ax³.

(b) If the set of palindromic polynomials in P₃ is denoted by S, then a basis for S is apparently { 1+x³ , x+x²} as every palindromic polynomial in P₃ can be expressed as a linear combination of 1+x³ and x+x². Then the subset of R⁴corresponding to S has a basis { (1,0,0,1), 0,1,1,0)} . The dimenson of this subspace is 2.

(c ) Let p₁(x)= a+ bx +bx² +ax³ and p₂(x) = c+dx+dx² + cx³ be two arbitrary elements of S. Then p₁(x)+ p₂(x) = (a+ bx +bx² +ax³) + (c+dx+dx² + cx³) = (a+c) + (b+d)x +(b+d)x² +(a+c)x³ It may be observed that p₁(x)+ p₂(x) is a pallindromic polynomial. Hence S is closed under vector addition.