Question

use pascals triangle to complete the expansion of $(r + s)^4$.
$r^4 + 4r^3s\square + 6r\square s^2 + 4rs^3 + s^4$

Explanation:

Step1: Recall Binomial Expansion

The binomial expansion of \((a + b)^n\) follows the pattern where the exponents of \(a\) decrease from \(n\) to \(0\) and the exponents of \(b\) increase from \(0\) to \(n\), with coefficients from Pascal's Triangle. For \((r + s)^4\), the general term is \(C(4, k)r^{4 - k}s^{k}\) where \(C(4, k)\) is the binomial coefficient.

Step2: Analyze the first blank (exponent of \(s\) in \(4r^3s^{\square}\))

In the binomial expansion, for the term \(4r^3s^{\square}\), since the exponent of \(r\) is \(3\) (and the total degree of the term should be \(4\) as it's from \((r + s)^4\)), the exponent of \(s\) should be \(4 - 3=1\). So the first blank is \(1\).

Step3: Analyze the second blank (exponent of \(r\) in \(6r^{\square}s^2\))

For the term \(6r^{\square}s^2\), the exponent of \(s\) is \(2\), so the exponent of \(r\) should be \(4 - 2 = 2\). So the second blank is \(2\).

Answer:

First blank: \(1\), Second blank: \(2\) (So the filled expansion is \(r^{4}+4r^{3}s^{1}+6r^{2}s^{2}+4rs^{3}+s^{4}\))