Question

use pascals triangle to complete the expansion of ((q - r)^6). (square q^{square} - 6q^5r + 15q^4r^2 - 20q^3r^3 + 15q^2r^4 - 6qr^5 + r^6)

Explanation:

Step1: Recall Pascal's Triangle for \(n = 6\)

The coefficients for \((a + b)^n\) using Pascal's Triangle for \(n = 6\) are \(1, 6, 15, 20, 15, 6, 1\). For \((q - r)^6\), we can rewrite it as \((q+(-r))^6\). The first term of the binomial expansion \((a + b)^n\) is \(a^n\) with coefficient 1. Here \(a = q\) and \(n = 6\), so the first term should be \(1\times q^6\).

Step2: Identify the missing parts

Looking at the given expansion, the first term has a coefficient (the first box) and an exponent for \(q\) (the second box). From the binomial expansion formula, the first term of \((q - r)^6\) is \(q^6\) with coefficient 1. So the first box (coefficient) is 1 and the second box (exponent of \(q\)) is 6.

Answer:

The first box (coefficient) is \(1\) and the second box (exponent of \(q\)) is \(6\), so the first term is \(1q^6\) (or \(q^6\)).