If one root of ` 10 x^3 -x^2 - 278 x + 165=0` is 5 then product of the remaining two roots is

Options

A. 33
B. `-33/5`
C. `-(33)/(10)`
D. `(33)/(10)`

Answer

To solve the problem, we need to find the product of the remaining two roots of the polynomial equation \( 10x^3 - x^2 - 278x + 165 = 0 \) given that one root is \( x = 5 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the known root**:
We have the polynomial:
\[ P(x) = 10x^3 - x^2 - 278x + 165 \]
and it is given that one root is \( x = 5 \).

2. **Use synthetic division to divide the polynomial by \( x - 5 \)**:
We will perform synthetic division of \( P(x) \) by \( x - 5 \).

- Coefficients of \( P(x) \): \( 10, -1, -278, 165 \)
- Set up synthetic division:
\[ \begin{array}{r|rrrr} 5 & 10 & -1 & -278 & 165 \\ & & 50 & 245 & -165 \\ \hline & 10 & 49 & -33 & 0 \\ \end{array} \]

The result of the synthetic division is:
\[ 10x^2 + 49x - 33 \]

3. **Set the quotient equal to zero**:
Now we need to find the roots of the quadratic equation:
\[ 10x^2 + 49x - 33 = 0 \]

4. **Use the quadratic formula**:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 10 \), \( b = 49 \), and \( c = -33 \).

- Calculate the discriminant:
\[ b^2 - 4ac = 49^2 - 4 \cdot 10 \cdot (-33) = 2401 + 1320 = 3721 \]

- Calculate the roots:
\[ x = \frac{-49 \pm \sqrt{3721}}{2 \cdot 10} = \frac{-49 \pm 61}{20} \]
This gives us two roots:
\[ x_1 = \frac{12}{20} = \frac{3}{5}, \quad x_2 = \frac{-110}{20} = -\frac{11}{2} \]

5. **Find the product of the remaining two roots**:
The product of the roots \( x_1 \) and \( x_2 \) can be calculated as:
\[ \text{Product} = x_1 \cdot x_2 = \left(\frac{3}{5}\right) \left(-\frac{11}{2}\right) = -\frac{33}{10} \]

### Final Answer:
The product of the remaining two roots is:
\[ -\frac{33}{10} \]