If roots of the equation `x^(4)-8x^(3)+bx^(2)+cx+16=0` are positive, then

Options

A. `b=8=c`
B. `b=-24,c=-32`
C. `b=24, c=-32`
D. `b=24, c=32`

Answer

To solve the problem, we need to analyze the polynomial equation given:

\[ x^4 - 8x^3 + bx^2 + cx + 16 = 0 \]

We know that the roots of this polynomial are positive. Let's denote the roots as \( \alpha_1, \alpha_2, \alpha_3, \alpha_4 \).

### Step 1: Use Vieta's Formulas

According to Vieta's formulas, we have:

1. The sum of the roots:
\[ \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 8 \]

2. The product of the roots:
\[ \alpha_1 \alpha_2 \alpha_3 \alpha_4 = 16 \]

### Step 2: Apply AM-GM Inequality

Since the roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:

\[ \frac{\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4}{4} \geq \sqrt[4]{\alpha_1 \alpha_2 \alpha_3 \alpha_4} \]

Substituting the known values:

\[ \frac{8}{4} \geq \sqrt[4]{16} \]

This simplifies to:

\[ 2 \geq 2 \]

### Step 3: Equality Condition in AM-GM

The equality condition in the AM-GM inequality holds when all the terms are equal. Therefore, we have:

\[ \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 \]

Let \( \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 = k \). Then:

\[ 4k = 8 \implies k = 2 \]

Thus, each root is \( 2 \).

### Step 4: Calculate Coefficients \( b \) and \( c \)

Now we can find \( b \) and \( c \):

1. **Finding \( c \)**:
The sum of the products of the roots taken three at a time (which gives us \( -c \)):
\[ \alpha_1 \alpha_2 \alpha_3 + \alpha_1 \alpha_2 \alpha_4 + \alpha_1 \alpha_3 \alpha_4 + \alpha_2 \alpha_3 \alpha_4 = 4 \cdot (2 \cdot 2 \cdot 2) = 4 \cdot 8 = 32 \]
Hence, \( -c = 32 \) or \( c = -32 \).

2. **Finding \( b \)**:
The sum of the products of the roots taken two at a time (which gives us \( b \)):
\[ \alpha_1 \alpha_2 + \alpha_1 \alpha_3 + \alpha_1 \alpha_4 + \alpha_2 \alpha_3 + \alpha_2 \alpha_4 + \alpha_3 \alpha_4 = 6 \cdot (2 \cdot 2) = 6 \cdot 4 = 24 \]
Thus, \( b = 24 \).

### Final Result

The values of \( b \) and \( c \) are:

\[ b = 24, \quad c = -32 \]