Question

use pascals triangle to complete the expansion of $(v + w)^{3}$. $v^{3}+v^{2}w + 3vw^{2}+w^{3}$

Explanation:

Step1: Recall Pascal's Triangle for power 3

Pascal's Triangle row for \(n = 3\) is \(1,3,3,1\).

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

For \((v+w)^3\), when \(k = 0\), \(\binom{3}{0}v^{3}w^{0}=1\times v^{3}\times1 = v^{3}\); when \(k = 1\), \(\binom{3}{1}v^{2}w^{1}=3\times v^{2}w\).

Answer:

The first blank is \(1\) and the second blank is \(3\). So the expansion is \(1v^{3}+3v^{2}w + 3vw^{2}+w^{3}\)