Question

use pascals triangle to complete the expansion of $(p - q)^3$. $p^3 + \square p^2q + \square pq^2 - q^3$

Explanation:

Step1: Recall Pascal's Triangle for exponent 3

Pascal's Triangle rows correspond to binomial exponents. For \(n = 3\), the row is \(1, 3, 3, 1\).

Step2: Analyze the binomial \((p - q)^3\)

The expansion of \((a + b)^n\) uses Pascal's coefficients. Here \(a = p\), \(b=-q\), \(n = 3\). The general term is \(\binom{3}{k}p^{3 - k}b^{k}\). For the \(p^2q\) term (\(k = 1\)), \(\binom{3}{1}=3\), but since \(b=-q\), the sign for \(k = 1\) is \(3\times p^{2}\times(-q)=- 3p^{2}q\)? Wait, no, wait the given form is \(p^{3}+\square p^{2}q+\square pq^{2}-q^{3}\). Wait, maybe I misread. Wait, \((p - q)^3=(p+(-q))^3\). So using Pascal's coefficients \(1,3,3,1\), the expansion is \(1\times p^{3}\times(-q)^{0}+3\times p^{2}\times(-q)^{1}+3\times p^{1}\times(-q)^{2}+1\times p^{0}\times(-q)^{3}\). Simplify each term: first term \(p^{3}\), second term \(3p^{2}(-q)=-3p^{2}q\), third term \(3p(-q)^{2}=3pq^{2}\), fourth term \((-q)^{3}=-q^{3}\). But the given expression is \(p^{3}+\square p^{2}q+\square pq^{2}-q^{3}\). Wait, that means the sign of the second term: in the given, it's \(+\square p^{2}q\), but our expansion has \(-3p^{2}q\). Wait, maybe there's a mistake in the problem's sign? Or maybe I messed up. Wait, no, let's re - expand \((p - q)^3\) directly: \((p - q)(p - q)(p - q)\). First multiply \((p - q)(p - q)=p^{2}-2pq + q^{2}\), then multiply by \((p - q)\): \(p^{3}-2p^{2}q+pq^{2}-p^{2}q + 2pq^{2}-q^{3}=p^{3}-3p^{2}q + 3pq^{2}-q^{3}\). Ah! So the expansion is \(p^{3}-3p^{2}q + 3pq^{2}-q^{3}\). But the problem's given form is \(p^{3}+\square p^{2}q+\square pq^{2}-q^{3}\). So to match the form, the coefficient of \(p^{2}q\) is \(-3\) (because \(p^{3}+(-3)p^{2}q + 3pq^{2}-q^{3}\)) and the coefficient of \(pq^{2}\) is \(3\). Wait, maybe the problem has a typo in the sign of the second term's operator? But according to the expansion, the coefficient of \(p^{2}q\) is \(-3\) and the coefficient of \(pq^{2}\) is \(3\). But the given expression has a '+' before the \(p^{2}q\) term. Wait, maybe the original problem was supposed to be \((p + q)^3\) but it's written as \((p - q)^3\)? No, the problem says \((p - q)^3\). Wait, let's check again. The expansion of \((p - q)^3=p^{3}-3p^{2}q + 3pq^{2}-q^{3}\). So in the given form \(p^{3}+\square p^{2}q+\square pq^{2}-q^{3}\), the \(\square\) for \(p^{2}q\) is \(-3\) and for \(pq^{2}\) is \(3\).

Answer:

The coefficient of \(p^{2}q\) is \(-3\) and the coefficient of \(pq^{2}\) is \(3\). So the first box (for \(p^{2}q\)) is \(-3\) and the second box (for \(pq^{2}\)) is \(3\).