Question

Use synthetic division to determine which of the following are the real zeroes of the function f left parenthesis x right parenthesis equals 3 x cubed plus 10 x squared plus 4 x minus 8.
(1 point)
Responses

negative 3 halves comma 2
Image with alt text: negative 3 halves comma 2

negative 2 comma 2 thirds
Image with alt text: negative 2 comma 2 thirds

negative 2 comma negative 2 thirds comma 2
Image with alt text: negative 2 comma negative 2 thirds comma 2

negative 2 comma 3 halves comma 2
Image with alt text: negative 2 comma 3 halves comma 2

Answer 1

To determine the real zeros of the function \( f(x) = 3x^3 + 10x^2 + 4x - 8 \) using synthetic division, we'll first test the given options one by one.

### Step 1: Test Possible Rational Roots

The Rational Root Theorem suggests that if there are any rational roots of the polynomial, they must be factors of the constant term (\(-8\)) divided by the factors of the leading coefficient (\(3\)). The possible rational roots are:

- Factors of \(-8\): \( \pm 1, \pm 2, \pm 4, \pm 8 \)
- Factors of \(3\): \( \pm 1, \pm 3 \)

Possible rational roots include:
\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}
\]

### Step 2: Test Each Candidate

Let's perform synthetic division for the candidates given in the responses.

#### 1. Testing \( x = 2 \)

Perform synthetic division using \( 2 \):

\[
\begin{array}{r|rrrr}
2 & 3 & 10 & 4 & -8 \\
& & 6 & 32 & 72 \\
\hline
& 3 & 16 & 36 & 64 \\
\end{array}
\]

Since the remainder is \( 64 \), \( x = 2 \) is **not a zero**.

#### 2. Testing \( x = -2 \)

Perform synthetic division using \( -2 \):

\[
\begin{array}{r|rrrr}
-2 & 3 & 10 & 4 & -8 \\
& & -6 & -8 & 8 \\
\hline
& 3 & 4 & -4 & 0 \\
\end{array}
\]

The remainder is \( 0 \). Therefore, \( x = -2 \) **is a zero**.

#### 3. Testing \( x = -\frac{2}{3} \)

Perform synthetic division using \( -\frac{2}{3} \):

\[
\begin{array}{r|rrrr}
-\frac{2}{3} & 3 & 10 & 4 & -8 \\
& & -2 & -\frac{16}{3} & 0 \\
\hline
& 3 & 8 & -\frac{8}{3} & -8 \\
\end{array}
\]

The remainder is not \( 0 \), so \( x = -\frac{2}{3} \) is **not a zero**.

### Summary of Results So Far

So far, we have found that:
- The polynomial has a zero at \( x = -2 \).
- \( x = 2 \) and \( x = -\frac{2}{3} \) are not zeros.

### Check for \( x = \frac{3}{2} \) and \( -\frac{3}{2} \)

#### Testing \( x = \frac{3}{2} \)

Perform synthetic division:

\[
\begin{array}{r|rrrr}
\frac{3}{2} & 3 & 10 & 4 & -8 \\
& & \frac{9}{2} & \frac{57}{2} & \frac{115}{2} \\
\hline
& 3 & 19 & 61 & 107 \\
\end{array}
\]

This is not a zero.

#### Testing \( x = -\frac{3}{2} \)

Perform synthetic division with \( -\frac{3}{2} \):

\[
\begin{array}{r|rrrr}
-\frac{3}{2} & 3 & 10 & 4 & -8 \\
& & -\frac{9}{2} & -\frac{3}{2} & 0 \\
\hline
& 3 & \frac{11}{2} & \frac{1}{2} & 0 \\
\end{array}
\]

Since the remainder is \( 0 \), \( x = -\frac{3}{2} \) is a zero.

### Final List of Real Zeros

The real zeros found through synthetic division are:
- \( x = -2 \)
- \( x = -\frac{3}{2} \)

Among the provided options, the only one that includes \( -2 \) is:
- **negative 2 comma negative 2 thirds comma 2**

Thus, the correct answer is:
\[
\text{negative 2, negative } \frac{2}{3}, \text{ 2}
\]