How do I use the intermediate value theorem to prove every polynomial of odd degree has at least one real root?
1 Answer
Given any polynomial
Explanation:
Let
Note that
If
To prove that:
Let
Note that
So we find:
#f(x_1) >= a_0x_1^n-(abs(a_1)x^(n-1)+abs(a_2)x^(n-2)+...+abs(a_n))#
#>= a_0x_1^n-(abs(a_1)x^(n-1)+abs(a_2)x^(n-1)+...+abs(a_n)x^(x-1))#
#= x^(n-1)(a_0x-(abs(a_1)+abs(a_2)+...+abs(a_n)))#
#= x^(n-1)((1+abs(a_0)+abs(a_1)+abs(a_2)+...+abs(a_n))-(abs(a_1)+abs(a_2)+...+abs(a_n)))#
#= x^(n-1)(1+abs(a_0)) > 0#
If
To prove that, note that if
We have:
#f(x)# continuous over#[-x_1, x_1]# #f(-x_1) < 0 < f(x_1)#
So by the intermediate value theorem:
#EE x in (-x_1, x_1) : f(x) = 0#
If the leading coefficient (