Question

1. expand the binomial $(x - 4)^4$ by applying pascal’s triangle.

options:

• $-4x^4 - 16x^3 + 96x^2 - 128x + 256$
• $x^4 - 16x^3 + 96x^2 - 256x + 256$
• $x^4 - 12x^3 + 12x^2 - 256x$
• $x^4 - 8x^3 + 48x^2 - 128x + 256$

Explanation:

Step 1: Recall Pascal's Triangle for \( n = 4 \)

For \( (a + b)^n \), the coefficients from Pascal's Triangle when \( n = 4 \) are \( 1, 4, 6, 4, 1 \).

Step 2: Identify \( a \) and \( b \) in the binomial

In \( (x - 4)^4 \), we have \( a = x \) and \( b = -4 \).

Step 3: Apply the binomial expansion formula

The binomial expansion is \( \sum_{k = 0}^{4} \binom{4}{k} a^{4 - k} b^{k} \), where \( \binom{4}{k} \) are the coefficients from Pascal's Triangle.

• For \( k = 0 \): \( \binom{4}{0} x^{4 - 0} (-4)^{0} = 1 \cdot x^4 \cdot 1 = x^4 \)
• For \( k = 1 \): \( \binom{4}{1} x^{4 - 1} (-4)^{1} = 4 \cdot x^3 \cdot (-4) = -16x^3 \)
• For \( k = 2 \): \( \binom{4}{2} x^{4 - 2} (-4)^{2} = 6 \cdot x^2 \cdot 16 = 96x^2 \)
• For \( k = 3 \): \( \binom{4}{3} x^{4 - 3} (-4)^{3} = 4 \cdot x \cdot (-64) = -256x \)
• For \( k = 4 \): \( \binom{4}{4} x^{4 - 4} (-4)^{4} = 1 \cdot 1 \cdot 256 = 256 \)

Step 4: Combine the terms

Adding these terms together: \( x^4 - 16x^3 + 96x^2 - 256x + 256 \)

Answer:

\( x^4 - 16x^3 + 96x^2 - 256x + 256 \) (the second option: \( x^4 - 16x^3 + 96x^2 - 256x + 256 \))