Question

use pascals triangle to complete the expansion of (y - z)^5. y^5 + -5 y^4 z^□ + 10y^3 z^2 - 10y^2 z^3 + 5yz^4 - z^5

Explanation:

Step1: Recall Binomial Expansion

The binomial expansion of \((a + b)^n\) is given by \(\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). For \((y - z)^5=(y+(-z))^5\), we use Pascal's Triangle for \(n = 5\). The coefficients from Pascal's Triangle for \(n = 5\) are \(1,5,10,10,5,1\).

Step2: Analyze the Term \(y^{4}z\)

In the term \(y^{4}z\), the power of \(y\) is \(4\) and the power of \(z\) (which is \(-z\) in the binomial) is \(1\). The general term is \(\binom{5}{k}y^{5 - k}(-z)^{k}\). For \(k = 1\), \(\binom{5}{1}=\frac{5!}{1!(5 - 1)!}=\frac{5!}{1!4!}=5\). But since \(b=-z\), the term is \(\binom{5}{1}y^{4}(-z)^{1}=- 5y^{4}z\). Also, the exponent of \(z\) here is \(1\) (because \(k = 1\), so the power of \(z\) is \(1\)).

Answer:

The missing exponent for \(z\) in the term \(-5y^{4}z^{\square}\) is \(1\), so the box should be filled with \(1\).