The expansion by the binomial theorem of `(2 x + (1)/(8) )^(10)` is `1024x^(10) + 640x^(9) + ax^(8) + bx^(7) + …` Calculate
(ii) coefficient of `x^(8)` in `(3x -2) ( 2x + (1)/(8) )^(10)`,

Answer

To solve the problem of finding the coefficient of \( x^8 \) in the expression \( (3x - 2)(2x + \frac{1}{8})^{10} \), we will follow these steps:

### Step 1: Find the coefficient of \( x^7 \) in \( (2x + \frac{1}{8})^{10} \)

The general term of the binomial expansion of \( (2x + \frac{1}{8})^{10} \) is given by:
\[ T_{r+1} = \binom{10}{r} (2x)^{10-r} \left(\frac{1}{8}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{10}{r} 2^{10-r} x^{10-r} \cdot \frac{1}{8^r} \]
\[ = \binom{10}{r} 2^{10-r} x^{10-r} \cdot 2^{-3r} \]
\[ = \binom{10}{r} 2^{10 - 4r} x^{10 - r} \]

To find the coefficient of \( x^7 \), we set \( 10 - r = 7 \) which gives \( r = 3 \).

Now substituting \( r = 3 \) into the general term:
\[ T_{4} = \binom{10}{3} 2^{10 - 4 \cdot 3} x^{7} \]
Calculating \( \binom{10}{3} \):
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Now substituting:
\[ T_{4} = 120 \cdot 2^{10 - 12} x^7 = 120 \cdot 2^{-2} x^7 = 120 \cdot \frac{1}{4} x^7 = 30 x^7 \]
Thus, the coefficient of \( x^7 \) is \( 30 \).

### Step 2: Find the coefficient of \( x^8 \) in \( (2x + \frac{1}{8})^{10} \)

Now, we need the coefficient of \( x^8 \). We set \( 10 - r = 8 \) which gives \( r = 2 \).

Substituting \( r = 2 \) into the general term:
\[ T_{3} = \binom{10}{2} 2^{10 - 4 \cdot 2} x^{8} \]
Calculating \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
Now substituting:
\[ T_{3} = 45 \cdot 2^{10 - 8} x^8 = 45 \cdot 2^{2} x^8 = 45 \cdot 4 x^8 = 180 x^8 \]
Thus, the coefficient of \( x^8 \) is \( 180 \).

### Step 3: Calculate the coefficient of \( x^8 \) in \( (3x - 2)(2x + \frac{1}{8})^{10} \)

Using the coefficients we found:
- Coefficient of \( x^7 \) is \( 30 \)
- Coefficient of \( x^8 \) is \( 180 \)

Now, we apply the expression:
\[ \text{Coefficient of } x^8 = 3 \cdot \text{(coefficient of } x^7) - 2 \cdot \text{(coefficient of } x^8) \]
Substituting the values:
\[ = 3 \cdot 30 - 2 \cdot 180 \]
Calculating:
\[ = 90 - 360 = -270 \]

### Final Answer:
The coefficient of \( x^8 \) in \( (3x - 2)(2x + \frac{1}{8})^{10} \) is \( -270 \).

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