Question

Pick an x-value between two zeroes. Solve the function at that value to determine if the graph is above or below the x-axis.
X^3 + 9x^2 + 23x + 15

Answer 1

To determine if the graph of the function \( f(x) = x^3 + 9x^2 + 23x + 15 \) is above or below the x-axis between two zeroes, we first need to find the zeroes of the function. This can often be done through numerical methods or graphing, but we can also use the Rational Root Theorem to test for potential rational zeroes.

### Step 1: Find the zeroes of the function

We can start testing for rational root candidates like \(\pm 1, \pm 3, \pm 5, \pm 15\). Let's evaluate \( f(x) \) at several values:

1. \( f(-1) = (-1)^3 + 9(-1)^2 + 23(-1) + 15 = -1 + 9 - 23 + 15 = 0 \)
- Thus, \( x = -1 \) is a root.

Now we can factor \( f(x) \) using synthetic division for \( x + 1 \):

\[
\begin{array}{r|rrrr}
-1 & 1 & 9 & 23 & 15 \\
& & -1 & -8 & -15 \\
\hline
& 1 & 8 & 15 & 0 \\
\end{array}
\]

This shows:
\[
f(x) = (x + 1)(x^2 + 8x + 15)
\]

Now we will try to factor \( x^2 + 8x + 15 \):
\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]
The complete factorization is:
\[
f(x) = (x + 1)(x + 3)(x + 5)
\]

### Step 2: Identify the zeroes

The zeroes of the function are:
- \( x = -1 \)
- \( x = -3 \)
- \( x = -5 \)

### Step 3: Choose an x-value between two zeroes

Between the zeroes \(-1\) and \(-3\), we can pick \( x = -2 \).

### Step 4: Evaluate the function at \( x = -2 \)

Now we evaluate \( f(-2) \):
\[
f(-2) = (-2)^3 + 9(-2)^2 + 23(-2) + 15
\]
Calculating each term:
- \( (-2)^3 = -8 \)
- \( 9(-2)^2 = 9(4) = 36 \)
- \( 23(-2) = -46 \)

Putting it together:
\[
f(-2) = -8 + 36 - 46 + 15 = -8 + 36 - 46 + 15 = -3
\]

### Conclusion:

Since \( f(-2) = -3 \), the function is below the x-axis at \( x = -2 \). Therefore, the graph of the function is below the x-axis in the interval between the zeroes \( -3 \) and \( -1 \).