To determine the possible numbers of positive and negative real zeros for the polynomial [tex]\( P(x) = 6x^4 + 4x^2 - 7x + 6 \)[/tex] using Descartes' Rule of Signs, follow these steps:
### (a) Possible number(s) of positive real zeros:
1. Identify coefficients and their respective signs for [tex]\( P(x) \)[/tex]:
The polynomial [tex]\( P(x) \)[/tex] is [tex]\( 6x^4 + 4x^2 - 7x + 6 \)[/tex].
The coefficients are: [tex]\( 6, 0, 4, -7, 6 \)[/tex].
2. Count the number of sign changes in the sequence of coefficients:
- [tex]\( 6 \)[/tex] to [tex]\( 0 \)[/tex]: No sign change.
- [tex]\( 0 \)[/tex] to [tex]\( 4 \)[/tex]: No sign change.
- [tex]\( 4 \)[/tex] to [tex]\( -7 \)[/tex]: Change from positive to negative (1st sign change).
- [tex]\( -7 \)[/tex] to [tex]\( 6 \)[/tex]: Change from negative to positive (2nd sign change).
There are 2 sign changes in the polynomial [tex]\( P(x) \)[/tex].
3. Possible numbers of positive real zeros:
According to Descartes' Rule of Signs, the number of positive real zeros equals the number of sign changes or any positive even number less than this amount.
Therefore, the possible numbers of positive real zeros are: 2 or 0.
### (b) Possible number(s) of negative real zeros:
1. Substitute [tex]\( x \)[/tex] with [tex]\(-x\)[/tex] in the polynomial to form [tex]\( P(-x) \)[/tex]:
[tex]\( P(-x) = 6(-x)^4 + 4(-x)^2 - 7(-x) + 6 \)[/tex]
= [tex]\( 6x^4 + 4x^2 + 7x + 6 \)[/tex].
2. Identify coefficients and their respective signs for [tex]\( P(-x) \)[/tex]:
The polynomial [tex]\( P(-x) \)[/tex] is [tex]\( 6x^4 + 4x^2 + 7x + 6 \)[/tex].
The coefficients are: [tex]\( 6, 0, 4, 7, 6 \)[/tex].
3. Count the number of sign changes in the sequence of coefficients:
- [tex]\( 6 \)[/tex] to [tex]\( 0 \)[/tex]: No sign change.
- [tex]\( 0 \)[/tex] to [tex]\( 4 \)[/tex]: No sign change.
- [tex]\( 4 \)[/tex] to [tex]\( 7 \)[/tex]: No sign change.
- [tex]\( 7 \)[/tex] to [tex]\( 6 \)[/tex]: No sign change.
There are no sign changes in the polynomial [tex]\( P(-x) \)[/tex].
4. Possible numbers of negative real zeros:
Since there are no sign changes, according to Descartes' Rule of Signs, the number of negative real zeros is 0.
Based on Descartes' Rule of Signs:
(a) Possible number(s) of positive real zeros: 2, 0
(b) Possible number(s) of negative real zeros: 0
