Homework $$ 1.5: $$ Zeros of Polynomial Functions Score: $$ 12.5 / 100 $$ Answered: $$ 2 / 10 $$ Question 3 Find all zeros of the function $$ f(x)=12 x^{3}-19 x^{2}-3 x+10 $$. Enter the zeros separated by commas. Question Help: $$ \square $$ Video Submit Question

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To find all the zeros of the polynomial function $$ f(x) = 12x^3 - 19x^2 - 3x + 10 $$, we can follow these steps:

### 1. **Use the Rational Root Theorem**

The Rational Root Theorem suggests that any rational root, expressed as $$ \frac{p}{q} $$, is such that:
- $$ p $$ is a factor of the constant term (10).
- $$ q $$ is a factor of the leading coefficient (12).

**Possible rational roots**:
$$
\pm1, \pm\frac{1}{2}, \pm\frac{1}{3}, \pm\frac{1}{4}, \pm\frac{1}{6}, \pm\frac{1}{12}, \pm2, \pm\frac{2}{3}, \pm5, \pm\frac{5}{2}, \pm\frac{5}{3}, \pm\frac{5}{4}, \pm\frac{5}{6}, \pm\frac{5}{12}, \pm10, \pm\frac{10}{3}, \ldots
$$

### 2. **Test Possible Roots**

Let's test $$ x = 1 $$:

$$
f(1) = 12(1)^3 - 19(1)^2 - 3(1) + 10 = 12 - 19 - 3 + 10 = 0
$$

Since $$ f(1) = 0 $$, $$ x = 1 $$ is a root.

### 3. **Polynomial Division**

Now, we can factor out $$ (x - 1) $$ from $$ f(x) $$:

$$
f(x) = (x - 1)(12x^2 - 7x - 10)
$$

### 4. **Find Zeros of the Quadratic Factor**

Solve the quadratic equation $$ 12x^2 - 7x - 10 = 0 $$ using the quadratic formula:

$$
x = \frac{7 \pm \sqrt{(-7)^2 - 4(12)(-10)}}{2(12)} = \frac{7 \pm \sqrt{49 + 480}}{24} = \frac{7 \pm \sqrt{529}}{24} = \frac{7 \pm 23}{24}
$$

**Solutions**:
- $$ x = \frac{7 + 23}{24} = \frac{30}{24} = \frac{5}{4} $$
- $$ x = \frac{7 - 23}{24} = \frac{-16}{24} = -\frac{2}{3} $$

### 5. **All Zeros of $$ f(x) $$**

Combining all the zeros, we have:
$$
x = 1, \quad x = \frac{5}{4}, \quad x = -\frac{2}{3}
$$

### **Final Answer**

$$
1,\ \frac{5}{4},\ -\frac{2}{3}
$$
The zeros of the function $$ f(x) = 12x^3 - 19x^2 - 3x + 10 $$ are $$ 1 $$, $$ \frac{5}{4} $$, and $$ -\frac{2}{3} $$.

Homework Zeros of Polynomial Functions
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Question 3
Find all zeros of the function . Enter the zeros separated by commas.
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Homework Zeros of Polynomial Functions Score: Answered: Question 3 Find all zeros of the function . Enter the zeros separated by commas. Question Help: Video Submit Question