To find the value of $k$ in the expression $(kx - 1 - \frac{7}{9}kx)(1 - 3x)^7$ where the coefficient of $x^2$ is $-147$, we need to expand the expression and collect terms involving $x^2$.

Step 1: Expand Each Part Separately

Consider the expression:

$(kx - 1 - \frac{7}{9}kx)(1 - 3x)^7$

This consists of two parts:

1. $(kx - 1 - \frac{7}{9}kx) = kx - 1 - \frac{7}{9}kx = \left(k - \frac{7}{9}k\right)x - 1 = \frac{2k}{9}x - 1$
2. The binomial expansion of $(1 - 3x)^7$.

Step 2: Binomial Expansion of $(1 - 3x)^7$

The binomial theorem states:

$(1 - 3x)^7 = \sum_{n=0}^{7} \binom{7}{n} (1)^{7-n}(-3x)^n = \sum_{n=0}^{7} \binom{7}{n} (-3x)^n$

Focusing on the terms involving $x$ and $x^2$:

• The term for $n = 1$ (the $x$ term) is: $\binom{7}{1}(-3x)^1 = 7(-3x) = -21x$

• The term for $n = 2$ (the $x^2$ term) is: $\binom{7}{2}(-3x)^2 = 21(9x^2) = 189x^2$

Step 3: Multiply and Collect $x^2$ Terms

Now we multiply the simplified first part $\left(\frac{2k}{9}x - 1\right)$ by the expansion:

$\left(\frac{2k}{9}x - 1\right)(1 - 21x + 189x^2 + \cdots)$

We are interested in the coefficient of $x^2$. The $x^2$ terms come from:

1. The product of $-1$ (constant term) and $189x^2$:
   $-1 \cdot 189x^2 = -189x^2$

2. The product of $\frac{2k}{9}x$ and $-21x$ (the $x$ term):
   $\frac{2k}{9}x \cdot (-21x) = \frac{2k}{9} \cdot (-21) x^2 = -\frac{42k}{9}x^2$

Step 4: Sum Up the Coefficients of $x^2$

The total coefficient of $x^2$ is:

$-189 - \frac{42k}{9}$

According to the problem, this coefficient is $-147$:

$-189 - \frac{42k}{9} = -147$

Step 5: Solve for $k$

1. Add 189 to both sides:

   $-\frac{42k}{9} = -147 + 189$

   $-\frac{42k}{9} = 42$

2. Multiply both sides by $-1$:

   $\frac{42k}{9} = -42$

3. Multiply both sides by 9:

   $42k = -42 \times 9$

   $42k = -378$

4. Divide both sides by 42:

   $k = \frac{-378}{42} = -9$

Answer

The value of $k$ is:

$\boxed{-9}$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you expand a binomial expression using the binomial theorem?
2. What are some common applications of the binomial theorem in probability and statistics?
3. How can we use the coefficients of binomial expansions to solve combinatorial problems?
4. What is the general formula for expanding a polynomial raised to a power?
5. How can we find the coefficient of a particular term in a polynomial expansion without expanding the entire expression?

Tip: Always carefully consider how terms multiply together when expanding polynomials, as mistakes in signs or coefficients are common.