Using the Descartes Rule of Sign count the number of sign changes of f(-x) and determine the number of negative real zeros.
1. f(x)=x^3+x^2-10x-24
My answer: A few signs due change but not enough to say they changed." 0 sign changes of f(-x), which means f(x) has 0 negative real zeros.
f(-x) = -x^3+x^2+10x-24
2 sign changes, mean either 2 or 0 negative roots
"A few" is not a very precise answer.
than you again steve! I know I should have said "2" I am sorry. Thank you for correcting me. I appreciate your help.
To determine the number of sign changes of f(-x) and thus the number of negative real zeros of f(x) using the Descartes' Rule of Signs, follow these steps:
Step 1: Write the equation for f(-x) by replacing x with -x:
f(-x) = (-x)^3 + (-x)^2 - 10(-x) - 24
f(-x) = -x^3 + x^2 + 10x - 24
Step 2: Count the number of sign changes in the coefficients of f(-x). A sign change occurs when there is a transition from a positive coefficient to a negative coefficient, or vice versa. Ignore any zero coefficients.
In the function f(-x) = -x^3 + x^2 + 10x - 24, there are two sign changes:
- The coefficient of x^3 changes from negative to positive.
- The coefficient of x changes from positive to negative.
Step 3: Determine the number of negative real zeros of f(x) by analyzing the sign changes from Step 2.
Since there are two sign changes in f(-x), the number of negative real zeros of f(x) can be 0, 2, or any even number. However, we cannot determine the exact number of negative real zeros without further analysis or using other methods.
Therefore, the statement "0 sign changes of f(-x), which means f(x) has 0 negative real zeros" is incorrect. Based on the given information, we can conclude that f(x) may have 0, 2, or any even number of negative real zeros.