Question

find the zeros of g(x)=4x^3+13x^2-37x-10 given its graph

Answer 1

To find the zeros of the polynomial function \( g(x) = 4x^3 + 13x^2 - 37x - 10 \) given its graph, we generally look for the points where the graph intersects the x-axis. These intersection points correspond to the values of \( x \) for which \( g(x) = 0 \).

However, without the visual of the graph, we can proceed by finding the roots analytically through methods like synthetic division or using the Rational Root Theorem to test possible rational roots.

### Step 1: Rational Root Theorem
According to the Rational Root Theorem, the possible rational roots are given by the factors of the constant term divided by the factors of the leading coefficient.

The constant term is \(-10\) and has the factors: \( \pm 1, \pm 2, \pm 5, \pm 10 \).
The leading coefficient is \(4\) and has the factors: \( \pm 1, \pm 2, \pm 4\).

Thus, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4}
\]

### Step 2: Testing Possible Roots
Let's test some of these possible roots using substitution:

1. Testing \( x = 1 \):
\[
g(1) = 4(1)^3 + 13(1)^2 - 37(1) - 10 = 4 + 13 - 37 - 10 = -30 \quad (\text{not a root})
\]

2. Testing \( x = -1 \):
\[
g(-1) = 4(-1)^3 + 13(-1)^2 - 37(-1) - 10 = -4 + 13 + 37 - 10 = 36 \quad (\text{not a root})
\]

3. Testing \( x = 2 \):
\[
g(2) = 4(2)^3 + 13(2)^2 - 37(2) - 10 = 32 + 52 - 74 - 10 = 0 \quad (\text{is a root})
\]

Now, we found that \( x = 2 \) is a root.

### Step 3: Synthetic Division
Now we can use synthetic division to simplify \( g(x) \) divided by \( x - 2 \).

```
2 | 4 13 -37 -10
| 8 42 10
----------------------
4 21 5 0
```
This gives us:
\[
g(x) = (x - 2)(4x^2 + 21x + 5)
\]

### Step 4: Finding Remaining Roots
Now we need to solve the quadratic \( 4x^2 + 21x + 5 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 4, b = 21, c = 5 \):
\[
x = \frac{-21 \pm \sqrt{21^2 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}
\]
\[
x = \frac{-21 \pm \sqrt{441 - 80}}{8}
\]
\[
x = \frac{-21 \pm \sqrt{361}}{8}
\]
\[
x = \frac{-21 \pm 19}{8}
\]

Calculating the two possibilities:
1. \( x = \frac{-2}{8} = -\frac{1}{4} \)
2. \( x = \frac{-40}{8} = -5 \)

### Conclusion
The zeros of \( g(x) = 4x^3 + 13x^2 - 37x - 10 \) are:
\[
x = 2, \quad x = -\frac{1}{4}, \quad x = -5
\]