Question

use pascals triangle to expand ((z + 1)^6). ((z + 1)^6 = square)

Explanation:

Step1: Get Pascal's Triangle row 6

The 6th row (starting from row 0) of Pascal's Triangle is $1, 6, 15, 20, 15, 6, 1$.

Step2: Apply binomial expansion formula

For $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$, here $a=z$, $b=1$, $n=6$.

$$\begin{align*} (z+1)^6 &= 1\cdot z^6\cdot1^0 + 6\cdot z^5\cdot1^1 + 15\cdot z^4\cdot1^2 + 20\cdot z^3\cdot1^3 + 15\cdot z^2\cdot1^4 + 6\cdot z^1\cdot1^5 + 1\cdot z^0\cdot1^6 \end{align*}$$

Step3: Simplify each term

Since $1^k=1$ and $z^0=1$, simplify each term:

$$ z^6 + 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 $$

Answer:

$z^6 + 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1$