Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
1. 9x^6-5x^4-2x^3+8x^2-4x=0
2. -9x^4+5x^3-2x^2+4x-3=0
To use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for a given polynomial function, follow these steps:
1. Count the number of sign changes in the function's coefficients.
- For the first function, 9x^6-5x^4-2x^3+8x^2-4x=0, the coefficients are (+)9, (-)5, (-)2, (+)8, (-)4. Counting the sign changes, we have (+ to -) 1 sign change.
2. Determine the maximum possible number of positive real zeros.
- The maximum possible number of positive real zeros is equal to the number of sign changes or a smaller even number. In this case, since there is 1 sign change, the maximum possible number of positive real zeros is 1.
3. Check the number of sign changes in f(-x).
- Replace x in the equation with -x and simplify. For the first function, f(-x) = 9(-x)^6-5(-x)^4-2(-x)^3+8(-x)^2-4(-x) = 9x^6-5x^4+2x^3+8x^2+4x. Counting the sign changes in f(-x), we have (+ to -) 0 sign changes.
4. Determine the maximum possible number of negative real zeros.
- The maximum possible number of negative real zeros is equal to the number of sign changes in f(-x) or a smaller even number. In this case, since there are 0 sign changes in f(-x), the maximum possible number of negative real zeros is 0.
Therefore, for the first function, there is 1 possible positive real zero and 0 possible negative real zeros.
Now let's apply Descartes' Rule of Signs to the second function:
1. Count the number of sign changes in the function's coefficients.
- For the second function, -9x^4+5x^3-2x^2+4x-3=0, the coefficients are (-)9, (+)5, (-)2, (+)4, (-)3. Counting the sign changes, we have (+ to -) 3 sign changes.
2. Determine the maximum possible number of positive real zeros.
- The maximum possible number of positive real zeros is equal to the number of sign changes or a smaller even number. In this case, since there are 3 sign changes, the maximum possible number of positive real zeros is 3 or 2 or 0.
3. Check the number of sign changes in f(-x).
- Replace x in the equation with -x and simplify. For the second function, f(-x) = -9(-x)^4+5(-x)^3-2(-x)^2+4(-x)-3 = -9x^4-5x^3-2x^2-4x-3. Counting the sign changes in f(-x), we have (+ to -) 1 sign change.
4. Determine the maximum possible number of negative real zeros.
- The maximum possible number of negative real zeros is equal to the number of sign changes in f(-x) or a smaller even number. In this case, since there is 1 sign change in f(-x), the maximum possible number of negative real zeros is 1.
Therefore, for the second function, there are 3 or 2 or 0 possible positive real zeros and 1 possible negative real zero.