This one has me stumped.
Find the least integral upper bound of the zeros of the function f(x)=x^3-x^2+1
So by the rational root theorem, 1 and -1 might be roots.
by using synthetic division, i get the following values:
for f(-1)=-1
f(0)=1
f(1)=1
f(2)=5
and there is changes in the signs of the remainder/quotient for -1 and 0, but no changes for 1 and 2.
So would 1 be the upper bound, or 2?
Thanks for the help.
after-thoughts:
the only real zero is negative, so I guess would 0 be the upper bound? maybe? lol
To find the least integral upper bound of the zeros of the function f(x) = x^3 - x^2 + 1, you have correctly applied the rational root theorem to identify potential roots. The rational root theorem states that if a polynomial with integer coefficients has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible root.
In this case, the constant term is 1 and the leading coefficient is 1, so the possible rational roots are the factors of 1. You correctly identified 1 and -1 as potential roots.
To determine whether these potential roots are also actual roots, you applied synthetic division. When performing synthetic division, you substitute the potential root into the polynomial and check for a remainder of 0. If the remainder is 0, it means the potential root is indeed a root of the polynomial.
From your calculations, you found that f(-1) = -1, f(0) = 1, f(1) = 1, and f(2) = 5. As you mentioned, there is a change in sign of the remainder when substituting -1 and 0, indicating that these are not roots. However, there is no change in sign when substituting 1 and 2, which suggests that they are potential candidates for actual roots.
To determine the least integral upper bound of the zeros of the function, you'll need to consider the values of f(x) for x > 2. Since you found that f(2) = 5, you can conclude that the upper bound for the zeros of the function is greater than 2.
Since you only performed synthetic division up to x = 2 and did not find a change in sign, it is not possible to definitively determine the exact upper bound without further analysis. However, based on the information you provided, you can conclude that the least integral upper bound of the zeros of the function f(x) = x^3 - x^2 + 1 is greater than 2.