(3z2 -4t3)2 = Blank 1z4 + Blank 2z2t3 + Blank 3t6

To solve the equation \((3z^2 - 4t^3)^2\) and express it in the form \( \text{Blank 1} \cdot z^4 + \text{Blank 2} \cdot z^2 t^3 + \text{Blank 3} \cdot t^6 \), we'll expand the square using the binomial expansion formula:

\[
(a - b)^2 = a^2 - 2ab + b^2
\]

Here, \( a = 3z^2 \) and \( b = 4t^3 \).

Expanding:

\[
(3z^2 - 4t^3)^2 = (3z^2)^2 - 2(3z^2)(4t^3) + (4t^3)^2
\]

Now, calculate each term:

1. \( (3z^2)^2 = 9z^4 \)
2. \( 2(3z^2)(4t^3) = 24z^2t^3 \)
3. \( (4t^3)^2 = 16t^6 \)

So, the expanded form is:

\[
9z^4 - 24z^2t^3 + 16t^6
\]

Thus:

- **Blank 1** = 9
- **Blank 2** = -24
- **Blank 3** = 16

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:

1. What is the general form of a binomial expansion?
2. How would the expansion differ if the expression were \( (3z^2 + 4t^3)^2 \)?
3. What happens if you change the exponents on \( z \) or \( t \)?
4. Can you factor the resulting polynomial \( 9z^4 - 24z^2t^3 + 16t^6 \)?
5. How does the distributive property apply when expanding polynomials?

**Tip:** Always verify each step in the binomial expansion, as small errors can lead to incorrect coefficients or powers.

Learn how to expand and solve the expression (3z^2 - 4t^3)^2 into the form 9z^4 - 24z^2t^3 + 16t^6. This problem involves applying the binomial expansion formula and understanding polynomial expansion techniques. Suitable for high school students studying algebra and binomial theorem.

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