Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function. $$ \begin{array}{ll}\text { 18) } f(x)=-8 x^{9}-5 x^{8}-2.5 x^{7}+7 x^{6}+x+10\end{array} $$

Question

UpStudy AI · Accepted Answer

To apply Descartes's Rule of Signs, we need to analyze the function $$ f(x) $$ for positive and negative real zeros.

### Step 1: Count the sign changes for $$ f(x) $$

The function is given as:
$$
f(x) = -8x^9 - 5x^8 - 2.5x^7 + 7x^6 + x + 10
$$

Let's list the coefficients:
- For $$ x^9 $$: $$-8$$ (negative)
- For $$ x^8 $$: $$-5$$ (negative)
- For $$ x^7 $$: $$-2.5$$ (negative)
- For $$ x^6 $$: $$7$$ (positive)
- For $$ x^5 $$: $$0$$ (not counted)
- For $$ x^4 $$: $$0$$ (not counted)
- For $$ x^3 $$: $$0$$ (not counted)
- For $$ x^2 $$: $$0$$ (not counted)
- For $$ x^1 $$: $$1$$ (positive)
- For the constant term: $$10$$ (positive)

Now, we count the sign changes:
- From $$-8$$ to $$-5$$: no change
- From $$-5$$ to $$-2.5$$: no change
- From $$-2.5$$ to $$7$$: **1 change** (negative to positive)
- From $$7$$ to $$1$$: no change
- From $$1$$ to $$10$$: no change

**Total sign changes for $$ f(x) $$: 1**

According to Descartes's Rule of Signs, the number of positive real zeros is equal to the number of sign changes or less than that by an even number. Therefore, the possible number of positive real zeros is:
- **1 positive real zero.**

### Step 2: Count the sign changes for $$ f(-x) $$

Next, we need to evaluate $$ f(-x) $$:
$$
f(-x) = -8(-x)^9 - 5(-x)^8 - 2.5(-x)^7 + 7(-x)^6 + (-x) + 10
$$
This simplifies to:
$$
f(-x) = 8x^9 - 5x^8 + 2.5x^7 + 7x^6 - x + 10
$$

Now, let's list the coefficients:
- For $$ x^9 $$: $$8$$ (positive)
- For $$ x^8 $$: $$-5$$ (negative)
- For $$ x^7 $$: $$2.5$$ (positive)
- For $$ x^6 $$: $$7$$ (positive)
- For $$ x^5 $$: $$0$$ (not counted)
- For $$ x^4 $$: $$0$$ (not counted)
- For $$ x^3 $$: $$0$$ (not counted)
- For $$ x^2 $$: $$0$$ (not counted)
- For $$ x^1 $$: $$-1$$ (negative)
- For the constant term: $$10$$ (positive)

Now, we count the sign changes:
- From $$8$$ to $$-5$$: **1 change** (positive to negative)
- From $$-5$$ to $$2.5$$: **1 change** (negative to positive)
- From $$2.5$$ to $$7$$: no change
- From $$7$$ to $$-1$$: **1 change** (positive to negative)
- From $$-1$$ to $$10$$: **1 change** (negative to positive)

**Total sign changes for $$ f(-x) $$: 4**

According to Descartes's Rule of Signs, the number of negative real zeros is equal to the number of sign changes or less than that by an even number. Therefore, the possible number of negative real zeros is:
- **4, 2, or 0 negative real zeros.**

### Summary of Results
- Possible number of positive real zeros: **1**
- Possible number of negative real zeros: **4, 2, or 0**
The possible number of positive real zeros is 1, and the possible number of negative real zeros is 4, 2, or 0.

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

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Calculus Questions

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