use pascals triangle to complete the expansion of $(q - r)^{4}$. $q^{4}+q^{3}r+q^{2}r^{2}+qr^{3}+r^{4}$

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Accepted Answer

# Explanation:
## Step1: Find the coefficients from Pascal's Triangle
For the power of $4$, the coefficients from Pascal's Triangle are $1,4,6,4,1$.
## Step2: Apply the binomial theorem with $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$, here $a = q$, $b=-r$, $n = 4$
$$
\begin{align*}
(q-r)^4&=\binom{4}{0}q^{4}(-r)^{0}+\binom{4}{1}q^{3}(-r)^{1}+\binom{4}{2}q^{2}(-r)^{2}+\binom{4}{3}q^{1}(-r)^{3}+\binom{4}{4}q^{0}(-r)^{4}\
\end{align*}
$$
$$
\begin{align*}
&=1\times q^{4}\times1+4\times q^{3}\times(-r)+6\times q^{2}\times r^{2}+4\times q\times(-r^{3})+1\times1\times r^{4}\
&=q^{4}- 4q^{3}r + 6q^{2}r^{2}-4qr^{3}+r^{4}
\end{align*}
$$

# Answer:
$1q^{4}+(- 4)q^{3}r + 6q^{2}r^{2}+(-4)qr^{3}+1r^{4}$

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Question

Explanation:

Step1: Find the coefficients from Pascal's Triangle

Step2: Apply the binomial theorem with \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), here \(a = q\), \(b=-r\), \(n = 4\)

Answer: