What does Descartes' rule of signs tell you about the possible number of positive real zeros and the possible number of negative real zeros of the function?

[tex]
f(y) = y^4 + 8y^3 - y + 15
[/tex]

The possible number of positive real zeros is [tex]$\square$[/tex], and the possible number of negative real zeros is [tex]$\square$[/tex].

(Use a comma to separate answers as needed.)



Answer :

To determine the possible number of positive and negative real zeros of the given polynomial [tex]\( f(y) = y^4 + 8y^3 - y + 15 \)[/tex], we use Descartes' Rule of Signs.

### Possible Number of Positive Real Zeros

1. Identify the coefficients of the polynomial:
The polynomial is [tex]\( y^4 + 8y^3 - y + 15 \)[/tex], and the coefficients, in order, are [tex]\( 1, 8, 0, -1, 15 \)[/tex].

2. Determine sign changes for positive zeros:
- Going from [tex]\( 1 \)[/tex] to [tex]\( 8 \)[/tex] (no sign change)
- Going from [tex]\( 8 \)[/tex] to [tex]\( 0 \)[/tex] (0 means it can be ignored for sign changes)
- Going from [tex]\( 0 \)[/tex] to [tex]\( -1 \)[/tex] (sign change)
- Going from [tex]\( -1 \)[/tex] to [tex]\( 15 \)[/tex] (sign change)

There is 1 sign change.

### Possible Number of Negative Real Zeros

1. Substitute [tex]\( y \)[/tex] with [tex]\( -y \)[/tex]:
To find the possible number of negative real zeros, we must substitute [tex]\( y \)[/tex] with [tex]\( -y \)[/tex] in the polynomial:
[tex]\[ f(-y) = (-y)^4 + 8(-y)^3 - (-y) + 15 = y^4 - 8y^3 + y + 15 \][/tex]

2. Identify the coefficients after substitution:
The new polynomial is [tex]\( y^4 - 8y^3 + y + 15 \)[/tex], and the coefficients, in order, are [tex]\( 1, -8, 0, 1, 15 \)[/tex].

3. Determine sign changes for negative zeros:
- Going from [tex]\( 1 \)[/tex] to [tex]\( -8 \)[/tex] (sign change)
- Going from [tex]\( -8 \)[/tex] to [tex]\( 0 \)[/tex] (0 means it can be ignored for sign changes)
- Going from [tex]\( 0 \)[/tex] to [tex]\( 1 \)[/tex] (sign change)
- Going from [tex]\( 1 \)[/tex] to [tex]\( 15 \)[/tex] (no sign change)

There is 1 sign change.

### Conclusion

According to Descartes' Rule of Signs:
- The possible number of positive real zeros is 1.
- The possible number of negative real zeros is 1.

Hence, the answer is:

The possible number of positive real zeros is [tex]\(1\)[/tex], and the possible number of negative real zeros is [tex]\(1\)[/tex].