Can you please check my work?
Use Decartes Rule of signs to determine the possible number of postive and negative real zeros for the given function.
1. f(x)=-7x^9+x^5-x^2+6
This is what I did:
There are 2 sign changes
f(-x)=-7(-x)^9+(-x)^5-(-x)^2+6
There are 2 sign changes
So, are there 2 or 0 positive zeros, 2 or 0 negative zeros?
2. f(x)10x^3-8x^2+x+5
There are 2 sign changes
f(-x)=10(-x)^3-8(-x)^2+(-x)+5
There are 2 sign changes
So, are there 2 or 0 positive zeros, no negative zeros?
correct
To check if your work is correct using Descartes' Rule of Signs, you need to count the number of sign changes in the original polynomial and also in its opposite.
1. For the function f(x) = -7x^9 + x^5 - x^2 + 6:
- In the original function, there are 2 sign changes: from negative to positive after -7x^9, and from positive to negative after x^5.
- In the function f(-x), you correctly changed the signs of the terms. There are still 2 sign changes: from positive to negative after -7(-x)^9, and from negative to positive after (-x)^5.
Based on Descartes' Rule of Signs:
- The number of positive zeros is either 2 or 0, since there are 2 sign changes in the original polynomial.
- The number of negative zeros is either 2 or 0, since there are 2 sign changes in the function f(-x).
So, your answer that there are "2 or 0 positive zeros, 2 or 0 negative zeros" is correct for the given function.
2. For the function f(x) = 10x^3 - 8x^2 + x + 5:
- In the original function, there are 2 sign changes: from positive to negative after 10x^3, and from negative to positive after -8x^2. Note that the constant term does not contribute to sign changes.
- In the function f(-x), you correctly changed the signs of the terms. There are still 2 sign changes: from negative to positive after 10(-x)^3, and from positive to negative after -8(-x)^2.
Based on Descartes' Rule of Signs:
- The number of positive zeros is either 2 or 0, since there are 2 sign changes in the original polynomial.
- There are no negative zeros, as there are no sign changes in the function f(-x).
So, your answer that there are "2 or 0 positive zeros, no negative zeros" is correct for the given function.
Overall, your work for both cases is correct.