Question

The coefficient of \( x^4 \) in the expansion of \( (1-2x)^5 \) is equal to:

• A. \( 40 \)
• B. \( 320 \)
• C. \( -320 \)
• D. \( -32 \)
• E. \( 80 \)

Hint:

When the first term of the binomial is 1, the expansion simplifies significantly as \( 1^{n-r} \) is always 1. Pay close attention to the sign of the second term; since the power \( r=4 \) is even, the negative sign in \( (-2x)^4 \) becomes positive.

Answer 1

Answer: (E)

Concept: The Binomial Theorem provides a method for expanding expressions of the form \( (a + b)^n \). The general term in such an expansion is given by the formula \( T_{r+1} = \binom{n}{r} a^{n-r} b^r \). To find the coefficient of a specific power of \( x \), we identify the value of \( r \) that results in that power and then calculate the numerical coefficient associated with that term.

Step 1: Identify the components of the binomial expression.
In the given expression \( (1 - 2x)^5 \), we identify the following parameters:
• The first term \( a = 1 \)
• The second term \( b = -2x \)
• The index or power \( n = 5 \) We are looking for the coefficient of \( x^4 \). In the general term \( \binom{5}{r} (1)^{5-r} (-2x)^r \), the power of \( x \) is determined by \( r \). Therefore, to find the term containing \( x^4 \), we must set \( r = 4 \).

Step 2: Apply the general term formula and simplify.
Using \( r = 4 \), we calculate the 5th term \( T_{4+1} \): \[ T_5 = \binom{5}{4} (1)^{5-4} (-2x)^4 \] We know that the binomial coefficient \( \binom{5}{4} = \frac{5!}{4!(5-4)!} = 5 \). Substituting this and simplifying the other parts of the expression: \[ T_5 = 5 \cdot (1)^1 \cdot (-2)^4 \cdot x^4 \] Since \( (-2)^4 = 16 \): \[ T_5 = 5 \cdot 1 \cdot 16 \cdot x^4 = 80x^4 \] The numerical coefficient of the term \( x^4 \) is \( 80 \).