If one root of `x^3 +3x^2 +5x +k=0` is sum of the other two roots then k=

Options

A. a) `11/4`
B. b) `17/6`
C. c) `25/9`
D. d) `33/8`

Answer

To solve the problem, we need to find the value of \( k \) in the equation \( x^3 + 3x^2 + 5x + k = 0 \) given that one root is the sum of the other two roots. Let's denote the roots of the polynomial as \( \alpha, \beta, \gamma \).

### Step-by-step Solution:

1. **Understanding the Relationship Between Roots:**
We know from the problem that one root, say \( \alpha \), is equal to the sum of the other two roots:
\[ \alpha = \beta + \gamma \]

2. **Using Vieta's Formulas:**
According to Vieta's formulas, for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} \)
- The sum of the product of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \)
- The product of the roots \( \alpha\beta\gamma = -\frac{d}{a} \)

For our equation \( x^3 + 3x^2 + 5x + k = 0 \):
- \( a = 1 \), \( b = 3 \), \( c = 5 \), and \( d = k \)

Therefore, we have:
\[ \alpha + \beta + \gamma = -3 \]

3. **Substituting \( \alpha \):**
Since \( \alpha = \beta + \gamma \), we can substitute this into the sum of the roots:
\[ \beta + \gamma + \beta + \gamma = -3 \]
This simplifies to:
\[ 2\beta + 2\gamma = -3 \]
Dividing the entire equation by 2 gives:
\[ \beta + \gamma = -\frac{3}{2} \]

4. **Finding \( \alpha \):**
Now substituting \( \beta + \gamma \) back into the equation for \( \alpha \):
\[ \alpha = \beta + \gamma = -\frac{3}{2} \]

5. **Calculating \( k \):**
Since \( \alpha \) is a root of the polynomial, we substitute \( \alpha = -\frac{3}{2} \) into the original equation:
\[ \left(-\frac{3}{2}\right)^3 + 3\left(-\frac{3}{2}\right)^2 + 5\left(-\frac{3}{2}\right) + k = 0 \]

Calculating each term:
- \( \left(-\frac{3}{2}\right)^3 = -\frac{27}{8} \)
- \( 3\left(-\frac{3}{2}\right)^2 = 3 \cdot \frac{9}{4} = \frac{27}{4} \)
- \( 5\left(-\frac{3}{2}\right) = -\frac{15}{2} \)

Now substituting these values into the equation:
\[ -\frac{27}{8} + \frac{27}{4} - \frac{15}{2} + k = 0 \]

To combine these fractions, convert them to have a common denominator of 8:
- \( \frac{27}{4} = \frac{54}{8} \)
- \( -\frac{15}{2} = -\frac{60}{8} \)

Thus, the equation becomes:
\[ -\frac{27}{8} + \frac{54}{8} - \frac{60}{8} + k = 0 \]
Simplifying this:
\[ \frac{-27 + 54 - 60}{8} + k = 0 \]
\[ \frac{-33}{8} + k = 0 \]
Therefore, solving for \( k \):
\[ k = \frac{33}{8} \]

### Final Answer:
\[ k = \frac{33}{8} \]