Question

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

Explanation:

Step1: Recall Pascal's Triangle for \(n = 3\)

Pascal's Triangle rows correspond to the binomial coefficients of \((x + y)^n\). For \(n = 3\), the row of Pascal's Triangle is \(1, 3, 3, 1\). These coefficients are used for the terms in the expansion of \((x + y)^3\) as \(1x^3y^0+3x^2y^1 + 3x^1y^2+1x^0y^3\).

Step2: Identify the coefficient for \(xy^2\)

In the given expansion \(x^3 + 3x^2y+\square xy^2 + y^3\), the coefficient of \(xy^2\) (which is the term with \(x^1y^2\)) should be the third coefficient in the row for \(n = 3\) in Pascal's Triangle, which is \(3\).

Answer:

\(3\)