Apply Descartes' Rule of Signs to determine the number of possible positive and negative real zeros for the given function.

[tex]v(x) = 7x^6 - 3x^4 - 3x^3 - 7x^2 + 4x - 6[/tex]

Each part may have more than one possibility. Use commas to separate multiple responses.

Part 1 of 2
Number of possible positive real zeros: [tex]\square[/tex]

Part 2 of 2
Number of possible negative real zeros: [tex]\square[/tex]



Answer :

To determine the number of possible positive and negative real zeros for the given function using Descartes' Rule of Signs, follow these steps:

### Given Function
The given polynomial function is:
[tex]\[ v(x) = 7x^6 - 3x^4 - 3x^3 - 7x^2 + 4x - 6 \][/tex]

### Part 1: Number of Possible Positive Real Zeros
Descartes' Rule of Signs states that the number of positive real zeros of a polynomial is either equal to the number of sign changes between consecutive non-zero coefficients or less than that by an even number.

1. Find the coefficients and observe the signs:
[tex]\[ 7, -3, -3, -7, 4, -6 \][/tex]

2. Count the number of sign changes:
- From [tex]\(7\)[/tex] to [tex]\(-3\)[/tex]: 1 sign change
- From [tex]\(-3\)[/tex] to [tex]\(-3\)[/tex]: no sign change
- From [tex]\(-3\)[/tex] to [tex]\(-7\)[/tex]: no sign change
- From [tex]\(-7\)[/tex] to [tex]\(4\)[/tex]: 1 sign change
- From [tex]\(4\)[/tex] to [tex]\(-6\)[/tex]: 1 sign change

Therefore, there are [tex]\(3\)[/tex] sign changes.

3. Possible number of positive real zeros:
According to Descartes' Rule of Signs, the number of possible positive real zeros is the number of sign changes or less by an even number. So, for [tex]\(3\)[/tex] sign changes, the possibilities are:
[tex]\[ 3,\, 1 \][/tex]

Thus, the number of possible positive real zeros is [tex]\(3\)[/tex] or [tex]\(1\)[/tex].

### Part 2: Number of Possible Negative Real Zeros
To determine the number of possible negative real zeros, consider [tex]\(v(-x)\)[/tex] and then apply the same rule.

1. Calculate [tex]\(v(-x)\)[/tex]:
[tex]\[ v(-x) = 7(-x)^6 - 3(-x)^4 - 3(-x)^3 - 7(-x)^2 + 4(-x) - 6 \][/tex]
Simplifying the exponents, we have:
[tex]\[ v(-x) = 7x^6 - 3x^4 + 3x^3 - 7x^2 - 4x - 6 \][/tex]

2. Find the coefficients and observe the signs:
[tex]\[ 7, -3, 3, -7, -4, -6 \][/tex]

3. Count the number of sign changes:
- From [tex]\(7\)[/tex] to [tex]\(-3\)[/tex]: 1 sign change
- From [tex]\(-3\)[/tex] to [tex]\(3\)[/tex]: 1 sign change
- From [tex]\(3\)[/tex] to [tex]\(-7\)[/tex]: 1 sign change
- From [tex]\(-7\)[/tex] to [tex]\(-4\)[/tex]: no sign change
- From [tex]\(-4\)[/tex] to [tex]\(-6\)[/tex]: no sign change

Therefore, there are [tex]\(3\)[/tex] sign changes.

4. Possible number of negative real zeros:
According to Descartes' Rule of Signs, the number of possible negative real zeros is the number of sign changes or less by an even number. So, for [tex]\(3\)[/tex] sign changes, the possibilities are:
[tex]\[ 3,\, 1 \][/tex]

Thus, the number of possible negative real zeros is [tex]\(3\)[/tex] or [tex]\(1\)[/tex].

### Summary
- Number of possible positive real zeros: [tex]\(3, 1\)[/tex]
- Number of possible negative real zeros: [tex]\(3, 1\)[/tex]