To expand $(3 - 2x^2)^4$ using Pascal’s Triangle, follow these steps:

Step 1: Identify the Binomial Expansion Formula

Using the Binomial Theorem: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ where:

• $a = 3$,
• $b = -2x^2$,
• $n = 4$.

Step 2: Find the Binomial Coefficients

From Pascal’s Triangle, the coefficients for $n = 4$ are: $1, 4, 6, 4, 1$

Step 3: Expand Using the Formula

Expanding $(3 - 2x^2)^4$:

$(3 - 2x^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (3)^{4-k} (-2x^2)^k$

Calculating each term:

1. For $k = 0$: $\binom{4}{0} (3)^4 (-2x^2)^0 = 1 \cdot 81 \cdot 1 = 81$

2. For $k = 1$: $\binom{4}{1} (3)^3 (-2x^2)^1 = 4 \cdot 27 \cdot (-2x^2) = -216x^2$

3. For $k = 2$: $\binom{4}{2} (3)^2 (-2x^2)^2 = 6 \cdot 9 \cdot 4x^4 = 216x^4$

4. For $k = 3$: $\binom{4}{3} (3)^1 (-2x^2)^3 = 4 \cdot 3 \cdot (-8x^6) = -96x^6$

5. For $k = 4$: $\binom{4}{4} (3)^0 (-2x^2)^4 = 1 \cdot 1 \cdot 16x^8 = 16x^8$

Step 4: Write the Final Answer

$81 - 216x^2 + 216x^4 - 96x^6 + 16x^8$

Final Answer:

$\mathbf{81 - 216x^2 + 216x^4 - 96x^6 + 16x^8}$

Would you like a breakdown of a specific step?

Here are some related questions:

1. How do you find binomial coefficients using Pascal’s Triangle?
2. What is the Binomial Theorem, and when is it used?
3. How does the sign of $b$ in $(a + b)^n$ affect the expansion?
4. Can Pascal’s Triangle be used for negative exponents?
5. How would the expansion change for $(3 + 2x^2)^4$?

Tip: Pascal’s Triangle helps quickly find coefficients without needing to compute factorials!