A polynomial is said to be palindromic if its coefficients are the same read lef

Question

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.

Explanation / Answer

2)

a) The P3 polynomial has coefficients (a,b,b,a) starting from zero degree to three diagram

A palindrome is same read left to right or right to left

abba is palindrome since if we reverse the number then also we will get the same answer abba

Hence the general polynomial in P3 will be of the form

a + bx + bx^2 + cx^3

b)

Since in this case, the second element must be equal to third element and 1st element must be euqual to fourth element

Hence the subset of R4 corresponding to S as span will be of the form (a1,a2,a3,a4), where a2=a3 and a1=a4

Since it is of the form (a1,a2,a2,a1), hence the dimension of this subspace is 2-Dimensional in R4, since it is having 2 unique elements

c) Set S is closed under vector addition

If P1(x) belongs to S and P2(x) belongs to S, then it implies P1(x) + P2(x) will belong to S

P1(x) = a1 + b1x + b1x^2 + a1x^3

P2(x) = a2 + b2x + b2x^2 + a2x^3

P1(x) + P2(x) = (a1+a2) + (b1+b2)x + (b1+b2)x^2 + (a1+a2)x^3

Since (a1+a2,b1+b2,b1+b2,a1+a2) is also a palindrome, hence S is closed under vector addition

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