Determine the term involving `x^3` in `( 5 - 2x)^6`

Answer

To determine the term involving \( x^3 \) in the expansion of \( (5 - 2x)^6 \), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

In our case, we have \( a = 5 \), \( b = -2x \), and \( n = 6 \).

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

Substituting our values:

\[ T_{r+1} = \binom{6}{r} (5)^{6-r} (-2x)^r \]

### Step 2: Find the Term Involving \( x^3 \)
We need the term where \( x^3 \) appears, which means we set \( r = 3 \):

\[ T_{4} = \binom{6}{3} (5)^{6-3} (-2x)^3 \]

### Step 3: Calculate the Binomial Coefficient
Calculate \( \binom{6}{3} \):

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Calculate the Powers
Now calculate \( (5)^{6-3} \) and \( (-2x)^3 \):

\[ (5)^{3} = 125 \]
\[ (-2x)^3 = -8x^3 \]

### Step 5: Substitute Back into the Term
Now substitute these values back into the term:

\[ T_{4} = 20 \cdot 125 \cdot (-8x^3) \]

### Step 6: Simplify the Expression
Now simplify:

\[ T_{4} = 20 \cdot 125 \cdot -8 x^3 \]
\[ = -20000 x^3 \]

### Final Answer
The term involving \( x^3 \) in the expansion of \( (5 - 2x)^6 \) is:

\[ -20000 x^3 \]