Let `(2x^(2) + 3x + 4)^(10) = sum_(r=0)^(20) a_(r) x^( r)` then the value of `(a_(8))/(a_(12))` is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the ratio \( \frac{a_8}{a_{12}} \) where \( (2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r \).

### Step-by-Step Solution

1. **Understanding the Expression**:
The expression given is \( (2x^2 + 3x + 4)^{10} \). We want to find the coefficients \( a_8 \) and \( a_{12} \) in the expansion of this polynomial.

2. **Using a Substitution**:
We can use a substitution to simplify our calculations. Let's replace \( x \) with \( \frac{2}{x} \) in the original expression:
\[ (2(\frac{2}{x})^2 + 3(\frac{2}{x}) + 4)^{10} \]
Simplifying this gives:
\[ (2 \cdot \frac{4}{x^2} + \frac{6}{x} + 4)^{10} = ( \frac{8}{x^2} + \frac{6}{x} + 4)^{10} \]

3. **Factoring Out \( x^{-2} \)**:
We can factor out \( x^{-2} \) from the expression:
\[ = x^{-20} (8 + 6x + 4x^2)^{10} \]
This means we can express the polynomial as:
\[ (8 + 6x + 4x^2)^{10} \cdot x^{-20} \]

4. **Setting Up the Coefficients**:
Now, we can express the coefficients. The original polynomial can be expressed as:
\[ \sum_{r=0}^{20} a_r x^r \]
After substitution, we have:
\[ \sum_{r=0}^{20} a_r \cdot 2^r \cdot x^{20-r} \]

5. **Finding Coefficients**:
To find \( a_8 \) and \( a_{12} \), we need to evaluate the coefficients of \( x^8 \) and \( x^{12} \) respectively:
- For \( a_8 \): Set \( r = 8 \)
- For \( a_{12} \): Set \( r = 12 \)

6. **Relating the Coefficients**:
From the equations, we have:
\[ 2^{10} a_8 = 2^{12} a_{12} \]
This simplifies to:
\[ \frac{a_8}{a_{12}} = \frac{2^{12}}{2^{10}} = 2^{2} = 4 \]

### Final Answer
Thus, the value of \( \frac{a_8}{a_{12}} \) is \( 4 \).