What is the degree of the polynomial of the following

Tyson Mcneil

Tyson Mcneil

Answered question

2022-04-19

What is the degree of the polynomial of the following expression ?
[x+(x31)12]5+[x(x31)12]5

Answer & Explanation

Coupewopmergorlpt

Coupewopmergorlpt

Beginner2022-04-20Added 13 answers

Usually, a polynomial is understood to have only non-negative powers of x (i.e. x0=1,x1=x,x2 etc.), so a-priori it's not even clear that this is a polynomial. However, if we open up the binomial theorem, we see that the odd powers cancel:
(x+y)n+(xy)n=k=0n(nk)xnk[yk+(y)k]
When k is odd yk+(y)k cancels. In your case, the term corresponding to k=2l has degree n2l+3l=n+l, and so the maximum is achieved for l=2 and n+l=7

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