Find the bounds of the zeros of the function \(f(x) = x^3 - 7x^2 + 14x - 8\)
Answer
Step 6: According to the Lower Bound Theorem, if a number b is plugged into the function and the result is negative, then -b is a lower bound on the zeros of the function. Plugging in the factors from step 3, we find that -8 is a lower bound.
Step 1 ：Step 1: Identify the leading coefficient and constant term. In this case, the leading coefficient is 1 and the constant term is -8.
Step 2 ：Step 2: Create a list of factors of the constant term. The factors of -8 are \([-1, 1, -2, 2, -4, 4, -8, 8]\).
Step 3 ：Step 3: Divide each factor by the leading coefficient. Since the leading coefficient is 1, this does not change the list of factors.
Step 4 ：Step 4: The factors from step 3 give the potential rational roots of the function. To find the bounds of the zeros, use the Upper Bound Theorem and the Lower Bound Theorem.
Step 5 ：Step 5: According to the Upper Bound Theorem, if a number b is plugged into the function and the result is positive, then b is an upper bound on the zeros of the function. Plugging in the factors from step 3, we find that 8 is an upper bound.
Step 6 ：Step 6: According to the Lower Bound Theorem, if a number b is plugged into the function and the result is negative, then -b is a lower bound on the zeros of the function. Plugging in the factors from step 3, we find that -8 is a lower bound.
