Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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

[tex]\[ f(x)=x^7+x^6+x^2+x+3 \][/tex]

A. 0 positive zeros, 2 or 0 negative zeros
B. 0 positive zeros, 1 negative zero
C. 0 positive zeros, 0 negative zeros
D. 0 positive zeros, 3 or 1 negative zeros

Sagot :

GinnyAnswer
To determine the possible number of positive and negative real zeros for the function [tex]\( f(x) = x^7 + x^6 + x^2 + x + 3 \)[/tex] using Descartes's Rule of Signs, follow these steps:

### Step 1: Determine the number of sign changes in [tex]\( f(x) \)[/tex]

To find the number of positive real zeros, examine the signs of the coefficients in the polynomial's terms:
[tex]\[ f(x) = x^7 + x^6 + x^2 + x + 3 \][/tex]

The coefficients are [1, 1, 0, 1, 1, 0, 1, 3]. Observe these coefficients:
- 1 (positive)
- 1 (positive)
- 0 (no change in sign)
- 1 (positive)
- 1 (positive)
- 0 (no change in sign)
- 1 (positive)
- 3 (positive)

Since there are no sign changes among the coefficients, there are 0 positive real zeros.

### Step 2: Determine the number of sign changes in [tex]\( f(-x) \)[/tex]

To find the number of negative real zeros, substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex] and then examine the signs of the coefficients:
[tex]\[ f(-x) = (-x)^7 + (-x)^6 + (-x)^2 + (-x) + 3 = -x^7 + x^6 + x^2 - x + 3 \][/tex]

So, the coefficients for [tex]\( f(-x) \)[/tex] are: [-1, 1, 0, -1, 1, 0, -1, 3]. Now, check the sign changes:
- -1 (negative)
- 1 (positive)
- 0 (no change in sign)
- -1 (negative)
- 1 (positive)
- 0 (no change in sign)
- -1 (negative)
- 3 (positive)

Count the sign changes:
1. From -1 to 1 (change of sign)
2. From 1 to -1 (change of sign)
3. From -1 to 1 (change of sign)
4. From 1 to -1 (change of sign)
5. From -1 to 3 (change of sign)

Thus, there are 5 sign changes in the coefficients of [tex]\( f(-x) \)[/tex].

### Step 3: Apply Descartes's Rule of Signs

According to Descartes's Rule of Signs:
- The number of positive real zeros is equal to the number of sign changes in [tex]\( f(x) \)[/tex] or less by an even integer.
- The number of negative real zeros is equal to the number of sign changes in [tex]\( f(-x) \)[/tex] or less by an even integer.

From our observations:
- There are 0 sign changes in [tex]\( f(x) \)[/tex], so there are 0 positive real zeros.
- There are 5 sign changes in [tex]\( f(-x) \)[/tex], which means the possible number of negative real zeros can be 5, 3, or 1 (since we subtract an even integer).

Out of the given answer choices, the correct one is:
D. 0 positive zeros, 3 or 1 negative zeros
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.