Find the zeros of the quadratic polynomial `3x^(2) -x-4` and verify the relationship between the zeros and the coefficient .

Answer

To find the zeros of the quadratic polynomial \(3x^2 - x - 4\) and verify the relationship between the zeros and the coefficients, we will follow these steps:

### Step 1: Write the quadratic equation
The given quadratic polynomial is:
\[ 3x^2 - x - 4 = 0 \]

### Step 2: Identify coefficients
From the standard form of a quadratic equation \(ax^2 + bx + c = 0\), we can identify:
- \(a = 3\)
- \(b = -1\)
- \(c = -4\)

### Step 3: Use the quadratic formula
To find the zeros, we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{1 + 48}}{6} \]
\[ x = \frac{1 \pm \sqrt{49}}{6} \]
\[ x = \frac{1 \pm 7}{6} \]

### Step 4: Calculate the zeros
Now, we will calculate the two possible values for \(x\):
1. Using the positive sign:
\[ x_1 = \frac{1 + 7}{6} = \frac{8}{6} = \frac{4}{3} \]
2. Using the negative sign:
\[ x_2 = \frac{1 - 7}{6} = \frac{-6}{6} = -1 \]

Thus, the zeros of the polynomial are:
\[ x_1 = \frac{4}{3}, \quad x_2 = -1 \]

### Step 5: Verify the relationship between zeros and coefficients
#### Sum of zeros
The sum of the zeros \(x_1 + x_2\) should equal \(-\frac{b}{a}\):
\[ x_1 + x_2 = \frac{4}{3} + (-1) = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]
Calculating \(-\frac{b}{a}\):
\[ -\frac{b}{a} = -\frac{-1}{3} = \frac{1}{3} \]

#### Product of zeros
The product of the zeros \(x_1 \cdot x_2\) should equal \(\frac{c}{a}\):
\[ x_1 \cdot x_2 = \frac{4}{3} \cdot (-1) = -\frac{4}{3} \]
Calculating \(\frac{c}{a}\):
\[ \frac{c}{a} = \frac{-4}{3} \]

### Conclusion
Both the sum and product of the zeros match the relationships derived from the coefficients:
- Sum of zeros: \(-\frac{b}{a} = \frac{1}{3}\)
- Product of zeros: \(\frac{c}{a} = -\frac{4}{3}\)

Thus, the relationship between the zeros and the coefficients is verified.