Question

use pascals triangle to complete the expansion of $(v + w)^{2}$. $v^{2}+vw+w^{2}$

Explanation:

Step1: Recall Pascal's Triangle for power 2

Pascal's Triangle row for \(n = 2\) is \(1\), \(2\), \(1\).

Step2: Apply binomial expansion formula \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\)

For \((v+w)^2\), when \(k = 0\), \(\binom{2}{0}v^{2-0}w^{0}=1\times v^{2}\times1 = v^{2}\); when \(k = 1\), \(\binom{2}{1}v^{2 - 1}w^{1}=2\times v\times w=2vw\); when \(k = 2\), \(\binom{2}{2}v^{2-2}w^{2}=1\times1\times w^{2}=w^{2}\)

Answer:

\(1v^{2}+2vw + 1w^{2}\)