Question

identify the coefficients of the expansion $(x - 2)^4$ using pascals triangle.
a. 1,2,1,2,1
b. 1,5,10,10,5,1
c. 1,3,3,1
d. 1,-8,24,-32,16

Explanation:

Answer:

The coefficients of the binomial expansion \((a + b)^n\) can be found using the \(n\) - th row of Pascal's triangle. For \(n = 4\), the 4 - th row of Pascal's triangle is \(1,4,6,4,1\).

The binomial expansion of \((x-2)^4=(x+( - 2))^4\) using the binomial theorem \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\) where \(a=x\), \(b=-2\) and \(n = 4\) is:

\[

$$\begin{align*} (x-2)^4&=\binom{4}{0}x^{4}(-2)^{0}+\binom{4}{1}x^{3}(-2)^{1}+\binom{4}{2}x^{2}(-2)^{2}+\binom{4}{3}x^{1}(-2)^{3}+\binom{4}{4}x^{0}(-2)^{4}\\ &=1\times x^{4}\times1+4\times x^{3}\times(- 2)+6\times x^{2}\times4 + 4\times x\times(-8)+1\times1\times16\\ &=x^{4}-8x^{3}+24x^{2}-32x + 16 \end{align*}$$

\]

The coefficients are \(1,-8,24,-32,16\)

So the answer is d. 1, -8, 24, -32, 16