Use Pascal s Triangle to expand the expression $( 2 x - 4 ) ^ { 5 }$ .
A) $2 x ^ { 5 } - 20 x ^ { 4 } + 160 x ^ { 3 } - 640 x ^ { 2 } + 1280 x - 1024$
B) $2 x ^ { 5 } - 5 x ^ { 4 } + 10 x ^ { 3 } - 10 x ^ { 2 } + 5 x - - 4$
C) $32 x ^ { 5 } - 80 x ^ { 4 } + 80 x ^ { 3 } - 40 x ^ { 2 } + 10 x - 1024$
D) $32 x ^ { 5 } - 1024$
E) $32 x ^ { 5 } - 320 x ^ { 4 } + 1280 x ^ { 3 } - 2560 x ^ { 2 } + 2560 x - 1024$

Question

Quizplus · Accepted Answer

To expand $(2x-4)^5$ using Pascal's triangle, we start by writing the first six rows of the triangle:

$$\begin{matrix}
& & & & 1 & & & & \
& & & 1 & & 1 & & & \
& & 1 & & 2 & & 1 & & \
& & 1 & & 3 & & 3 & & 1 \
& & 1& & 4 & & 6 & & 4 & & 1 \
& & 1 & & 5 & & 10 & & 10 & & 5 & & 1 \
\end{matrix}$$

The coefficients of the expanded expression will be the entries in the 6th row of Pascal's triangle: 1, 5, 10, 10, 5, 1. We will also have powers of $2x$ and powers of $-4$, starting with 5 and decreasing to 0.

Using the row of coefficients and powers, we can build the expanded expression term by term. The first term is:

$$(2x)^5 = 32x^5$$

The second term is:

$$\binom{5}{1}(2x)^4(-4)^1 = -320x^4$$

The third term is:

$$\binom{5}{2}(2x)^3(-4)^2 = 1280x^3$$

The fourth term is:

$$\binom{5}{3}(2x)^2(-4)^3 = -2560x^2$$

The fifth term is:

$$\binom{5}{4}(2x)^1(-4)^4 = 2560x$$

The final term is:

$$\binom{5}{5}(2x)^0(-4)^5 = -1024$$

Summing these terms, we get:

$$32x^5 - 320x^4 + 1280x^3 - 2560x^2 + 2560x - 1024$$

Therefore, the answer is E.

Use Pascal S Triangle to Expand the Expression (2x−4)5( 2 x - 4 ) ^ { 5 }(2x−4)5

Use Pascal S Triangle to Expand the Expression $( 2 x - 4 ) ^ { 5 }$