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 equation 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, for a polynomial of the form \( ax^n + bx^{n-1} + cx^{n-2} + ... + k = 0 \):

1. The sum of the roots \( \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = -\frac{b}{a} \)
2. The product of the roots \( \alpha_1 \alpha_2 \alpha_3 \alpha_4 = \frac{(-1)^n k}{a} \)

For our polynomial, \( a = 1 \) and \( k = 16 \).

### Step 2: Calculate the Sum of the Roots

From Vieta's, we have:

\[ \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 8 \]

This means that the sum of the roots is positive.

### Step 3: Calculate the Product of the Roots

Using Vieta's again, we find the product:

\[ \alpha_1 \alpha_2 \alpha_3 \alpha_4 = 16 \]

Since all roots are positive, this product is also positive.

### Step 4: Apply the AM-GM Inequality

Since all 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:

\[ 2 \geq 2 \]

This shows that equality holds, which implies that all roots are equal.

### Step 5: Conclude the Roots

Since the roots are equal and their sum is 8, we can conclude:

\[ \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 = 2 \]

### Step 6: Determine \( b \) and \( c \)

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

1. The sum of the products of the roots taken two at a time (which gives us \( b \)):
\[ b = \alpha_1 \alpha_2 + \alpha_1 \alpha_3 + \alpha_1 \alpha_4 + \alpha_2 \alpha_3 + \alpha_2 \alpha_4 + \alpha_3 \alpha_4 \]
Since all roots are 2:
\[ b = 6 \cdot (2 \cdot 2) = 24 \]

2. The sum of the products of the roots taken three at a time (which gives us \( c \)):
\[ c = \alpha_1 \alpha_2 \alpha_3 + \alpha_1 \alpha_2 \alpha_4 + \alpha_1 \alpha_3 \alpha_4 + \alpha_2 \alpha_3 \alpha_4 \]
Again, since all roots are 2:
\[ c = 4 \cdot (2 \cdot 2 \cdot 2) = 32 \]

### Final Answer

Thus, the values of \( b \) and \( c \) are:

- \( b = 24 \)
- \( c = 32 \)