Question

Given
f
(
3
)
=
0
f(3)=0 of
f
(
x
)
=
x
3
+
5
x
2
+
8
x
−
96
f(x)=x
3
+5x
2
+8x−96, find all of the zeros. Separate your answers with a comma and use
±
± when needed. Remember to include all zeros.

Answer 1

To find the zeros of the polynomial \( f(x) = x^3 + 5x^2 + 8x - 96 \), we know that one of the zeros is \( x = 3 \) (since \( f(3) = 0 \)). We can use synthetic division to factor the polynomial \( f(x) \) by \( x - 3 \).

### Synthetic Division
1. Set up synthetic division with \( 3 \):
- Coefficients of \( f(x) \): \( 1, 5, 8, -96 \)

```
3 | 1 5 8 -96
| 3 24 96
---------------------
1 8 32 0
```

The result is \( x^2 + 8x + 32 \).

### Finding the Zeros of \( x^2 + 8x + 32 \)
Next, we set the quadratic to zero and solve for \( x \):

\[
x^2 + 8x + 32 = 0
\]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1 \), \( b = 8 \), and \( c = 32 \).

1. Calculate the discriminant:
\[
b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 32 = 64 - 128 = -64
\]

2. Since the discriminant is negative, we have two complex roots:
\[
x = \frac{-8 \pm \sqrt{-64}}{2 \cdot 1}
\]
\[
= \frac{-8 \pm 8i}{2}
\]
\[
= -4 \pm 4i
\]

### Summary of Zeros
Thus, the zeros of the polynomial \( f(x) = x^3 + 5x^2 + 8x - 96 \) are:
\[
\boxed{3, -4 + 4i, -4 - 4i}
\]