Question

use pascals triangle to complete the expansion of ((q - r)^3). (square q^3 - 3q^2r + 3qr^2 + square r^3)

Explanation:

Step1: Recall Pascal's Triangle for exponent 3

For \((a + b)^n\), the coefficients from Pascal's Triangle for \(n = 3\) are \(1, 3, 3, 1\). But here we have \((q - r)^3=(q+(-r))^3\).

Step2: Determine the first coefficient

The first term of the expansion \((q - r)^3\) is \(1\times q^3\), so the first box is \(1\).

Step3: Determine the last coefficient

The last term of the expansion \((q - r)^3\) is \(1\times(-r)^3=-r^3\), so the coefficient of \(r^3\) is \(- 1\).

Answer:

First box: \(1\), Second box: \(-1\)

So the expansion is \(1q^{3}-3q^{2}r + 3qr^{2}+(-1)r^{3}\) or \(q^{3}-3q^{2}r + 3qr^{2}-r^{3}\)