If roots of the equation `f(x)=x^6-12 x^5+bx^4+cx^3+dx^2+ex+64=0`are positive, then
remainder when `f(x)` is divided by `x-1` is (a) 2 (b) 1 (c) 3 (d) 10

Options

A. 2
B. 1
C. 3
D. 10

Explanation

To solve the problem, we need to find the remainder when the polynomial \( f(x) = x^6 - 12x^5 + bx^4 + cx^3 + dx^2 + ex + 64 \) is divided by \( x - 1 \). Given that all roots of the polynomial are positive, we can utilize the properties of the roots and the relationship between the coefficients and the roots.

### Step-by-step Solution:

1. **Identify the Sum and Product of Roots**:
According to Vieta's formulas, for a polynomial of the form \( ax^n + bx^{n-1} + ... + k = 0 \):
- The sum of the roots \( (x_1 + x_2 + x_3 + x_4 + x_5 + x_6) \) is given by \( -\frac{\text{coefficient of } x^{n-1}}{\text{coefficient of } x^n} \).
- The product of the roots \( (x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5 \cdot x_6) \) is given by \( (-1)^n \cdot \frac{\text{constant term}}{\text{coefficient of } x^n} \).

For our polynomial:
- The sum of the roots is \( 12 \) (since \( -(-12)/1 = 12 \)).
- The product of the roots is \( 64 \) (since \( 64/1 = 64 \)).

2. **Apply the AM-GM Inequality**:
Since all roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq (x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5 \cdot x_6)^{1/6} \]
Substituting the values we found:
\[ \frac{12}{6} \geq 64^{1/6} \]
Simplifying gives:
\[ 2 \geq 2 \]
This indicates that the equality holds, which occurs only when all the roots are equal.

3. **Determine the Roots**:
Since the equality holds in AM-GM, we conclude that:
\[ x_1 = x_2 = x_3 = x_4 = x_5 = x_6 = 2 \]

4. **Express the Polynomial**:
The polynomial can be expressed as:
\[ f(x) = (x - 2)^6 \]

5. **Find the Remainder**:
To find the remainder when \( f(x) \) is divided by \( x - 1 \), we evaluate \( f(1) \):
\[ f(1) = (1 - 2)^6 = (-1)^6 = 1 \]

Thus, the remainder when \( f(x) \) is divided by \( x - 1 \) is \( 1 \).

### Final Answer:
The remainder is \( 1 \), which corresponds to option (b).

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