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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].
### 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].
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