Question

use pascals triangle to complete the expansion of $(x + y)^{3}$. $x^{3}+x^{2}y+xy^{2}+y^{3}$

Explanation:

Step1: Recall Pascal's Triangle row for power 3

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

Step2: Apply the coefficients to the expansion

For \((x + y)^3\), using the binomial expansion formula \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), here \(a=x\), \(b = y\), \(n = 3\).
The terms are:

• When \(k = 0\): \(\binom{3}{0}x^{3}y^{0}=1\times x^{3}\)
• When \(k = 1\): \(\binom{3}{1}x^{2}y^{1}=3\times x^{2}y\)
• When \(k = 2\): \(\binom{3}{2}x^{1}y^{2}=3\times xy^{2}\)
• When \(k = 3\): \(\binom{3}{3}x^{0}y^{3}=1\times y^{3}\)

Answer:

\(1x^{3}+3x^{2}y + 3xy^{2}+1y^{3}\)