Question

use pascals triangle to complete the expansion of $(u + v)^4$. $u^{4}+u^{3}v+u^{2}v^{2}+uv^{3}+v^{4}$

Explanation:

Step1: Recall Pascal's Triangle row for power 4

Pascal's Triangle rows: Row 0: \(1\); Row 1: \(1\ 1\); Row 2: \(1\ 2\ 1\); Row 3: \(1\ 3\ 3\ 1\); Row 4: \(1\ 4\ 6\ 4\ 1\)

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

For \((u + v)^4\), \(n = 4\). The coefficients from Pascal's Triangle (Row 4) are \(1,4,6,4,1\)
So \((u + v)^4=1u^{4}+4u^{3}v + 6u^{2}v^{2}+4uv^{3}+1v^{4}\)

Answer:

\(1u^{4}+4u^{3}v + 6u^{2}v^{2}+4uv^{3}+1v^{4}\)