Question

1. Name the polynomial x3-1 = P(x)​

Answer 1

Step-by-step explanation:

I want to expand upon an idea expressed in the prior answer

The idea of:

xn−1x−1

=n∑r=1xn−r

or not in sigma notation:

xn−1x−1=xn−1+xn−2+...+x+1

We can prove this via induction:

Basis case :

⇒n=1

LHS:

x1−1x−1=1

RHS:

x1−1=x0=1

Hence basis case holds

Induction:

Assume

n=k

holds:

xk−1x−1=k∑r

=1xk−r

n=k+1 :

k+1∑r=1

xk+1−r=(k∑r=1 xk+1−r)+1

=x⋅(k∑r=1xk−r)+1

=x⋅(xk−1x−1)+1

=xk+1−x x−1+1

=xk+1−xx−1+x−1

x−1

=xk+1−1x−1

Hence this is also what we yield when plugging directly into formula:

Hence holds for all

k∈Z+

and all

k+1∈Z+

so holds for all

n∈Z+

⇒

Proven by mathematical induction

I thought this was a nice idea to consider!