Use Pascal's Triangle to expand $(z+1)^6$.

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03-11-2026
Mathematics

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Use Pascal's Triangle to expand $(z+1)^6$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T12:58:32-05:00

Apply the Binomial Theorem to identify the expansion formula: $(z+1)^6 = \sum_{k=0}^{6} \binom{6}{k} z^{6-k}$.
Use Pascal's Triangle to determine the coefficients for the 6th row: 1, 6, 15, 20, 15, 6, 1.
Substitute these coefficients back into the expansion formula.
The final expansion is $\boxed{z^6 + 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1}$.

Explanation

Understanding the Binomial Theorem

To expand the expression $(z+1)^6$, we use the Binomial Theorem. The theorem states that for any non-negative integer $n$, the expansion of $(z+w)^n$ is given by:
\[(z+w)^n = \sum_{k=0}^{n} \binom{n}{k} z^{n-k} w^k\]
In our case, $n=6$, $z=z$, and $w=1$. Therefore, the expression becomes:
\[(z+1)^6 = \sum_{k=0}^{6} \binom{6}{k} z^{6-k} 1^k = \sum_{k=0}^{6} \binom{6}{k} z^{6-k}\]

Identifying Coefficients Using Pascal's Triangle

Pascal's triangle is a useful tool to determine the binomial coefficients $\binom{n}{k}$. Each row $n$ of the triangle contains the coefficients for the binomial expansion of power $n$. We generate the rows as follows:

Row 0: 1
Row 1: 1, 1
Row 2: 1, 2, 1
Row 3: 1, 3, 3, 1
Row 4: 1, 4, 6, 4, 1
Row 5: 1, 5, 10, 10, 5, 1
Row 6: 1, 6, 15, 20, 15, 6, 1

The coefficients for $n=6$ are 1, 6, 15, 20, 15, 6, and 1.

Performing the Expansion

Now, we substitute these coefficients into the binomial expansion formula:
\[(z+1)^6 = 1 \cdot z^6 + 6 \cdot z^5 + 15 \cdot z^4 + 20 \cdot z^3 + 15 \cdot z^2 + 6 \cdot z + 1\]
Simplifying this, we get the expanded polynomial:
\[(z+1)^6 = z^6 + 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1\]
The final expansion is $\boxed{z^6 + 6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1}$.

Examples

The Binomial Theorem is used in various fields, such as probability, statistics, and computer science. For instance, in probability, if you flip a coin 6 times, the binomial coefficients help you determine the number of ways to get exactly $k$ heads, which is $\binom{6}{k}$. Understanding these expansions allows you to quickly calculate outcomes of multiple independent events or model growth and decline patterns in real-world phenomena.