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: \[ \text{negative 2, negative } \frac{2}{3}, \text{ 2} \]