Let $$ r $$ and $$ C $$ be real numbers. If $$ r $$ and $$ -r $$ are roots of
$$ 5x^3 + Cx^2 - 10x + 9 = 0 $$,
then $$ C $$ equals:

Question

Let $$ r $$ and $$ C $$ be real numbers. If $$ r $$ and $$ -r $$ are roots of  
$$ 5x^3 + Cx^2 - 10x + 9 = 0 $$,  
then $$ C $$ equals:

Accepted Answer

To solve this problem:

1. **Given Roots**:  
   Since $$ r $$ and $$ -r $$ are roots of the cubic equation, the third root must be some constant, say $$ k $$. The polynomial can be expressed as:  
   $$
   (x - r)(x + r)(x - k) = 0.
   $$
   Expanding this:  
   $$
   (x^2 - r^2)(x - k) = x^3 - kx^2 - r^2x + kr^2.
   $$

2. **Compare with the Given Equation**:  
   The given equation is:  
   $$
   5x^3 + Cx^2 - 10x + 9 = 0.
   $$
   Multiply the expanded form by 5 to match the coefficient of $$ x^3 $$:  
   $$
   5(x^3 - kx^2 - r^2x + kr^2) = 5x^3 - 5kx^2 - 5r^2x + 5kr^2.
   $$
   Comparing coefficients:  
   - Coefficient of $$ x^2: C = -5k $$.

3. **Determine Roots**:  
   From the given equation, substitute one root into the polynomial to find relationships between coefficients. Using symmetry and substituting values leads to:  
   $$
   r = \pm \sqrt{2}.
   $$
   The third root, $$ k $$, satisfies:  
   $$
   k = \frac{-9}{5}.
   $$
   
4. **Calculate Coefficient of $$ x^2 $$**:  
   Substitute into the expression for $$ C $$:  
   $$
   C = -5k = -5 	imes \frac{-9}{5} = -\frac{9}{2}.
   $$

Thus, the correct answer is **$$ A. -\frac{9}{2} $$**.

Answer

Answer

Answer