Determine the number of possible positive and negative real zeros for the given function. $$ p(x)=8 x^{7}+2 x^{4}-8 x^{3}+6 x^{2}-6 x+7 $$ Part: 0 / 3 Part 1 of 3 $$ p(x) $$ has real coefficients and the constant term is nonzero, so we can apply Descartes' Rule of Signs. To determine the number of possible positive real zeros, determine the number of sign changes in $$ p(x) $$. $$ p(x) $$ has $$ \square $$ sign change(s). The number of possible positive real zeros is either (Choose one) $$
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Determine the number of possible positive and negative real zeros for the given function. $$ p(x)=8 x^{7}+2 x^{4}-8 x^{3}+6 x^{2}-6 x+7 $$ Part: 0 / 3 Part 1 of 3 $$ p(x) $$ has real coefficients and the constant term is nonzero, so we can apply Descartes' Rule of Signs. To determine the number of possible positive real zeros, determine the number of sign changes in $$ p(x) $$. $$ p(x) $$ has $$ \square $$ sign change(s). The number of possible positive real zeros is either (Choose one) $$ 
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$$ p(x)=8x^7+2x^4-8x^3+6x^2-6x+7 $$

1. Identify the coefficients in order of descending degree:

- $$8$$ (from $$8x^7$$)
   - $$2$$ (from $$2x^4$$)
   - $$-8$$ (from $$-8x^3$$)
   - $$6$$ (from $$6x^2$$)
   - $$-6$$ (from $$-6x$$)
   - $$7$$ (constant term)

2. Count the sign changes:

- $$8$$ to $$2$$: no change (both are positive)
   - $$2$$ to $$-8$$: one sign change (positive to negative)
   - $$-8$$ to $$6$$: one sign change (negative to positive)
   - $$6$$ to $$-6$$: one sign change (positive to negative)
   - $$-6$$ to $$7$$: one sign change (negative to positive)

Total number of sign changes is $$4$$.

3. By Descartes' Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even integer. Thus, the possible numbers of positive real zeros are:

$$
   4, \quad 4-2=2, \quad \text{or} \quad 4-4=0.
   $$

So, the answer is:

$$ p(x) $$ has $$4$$ sign change(s), and the number of possible positive real zeros is either $$4$$, $$2$$, or $$0$$.
$$ p(x) $$ has 4 sign changes, so there are either 4, 2, or 0 positive real zeros.

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