-12t^7 18t^9 over - 6t^3 simplified is? -3t^12 - 12t^7, - 2t^3 - 3t, - 3t^6(4t-1), t^4(2-3t^2). Which answer is correct

Let's simplify the given expression step by step:

The expression is:

\[
\frac{-12t^7 + 18t^9}{-6t^3}
\]

### Step 1: Simplify each term by dividing by \(-6t^3\).

For the first term, \(\frac{-12t^7}{-6t^3}\):

\[
\frac{-12}{-6} = 2 \quad \text{and} \quad \frac{t^7}{t^3} = t^{7-3} = t^4
\]

So, \(\frac{-12t^7}{-6t^3} = 2t^4\).

For the second term, \(\frac{18t^9}{-6t^3}\):

\[
\frac{18}{-6} = -3 \quad \text{and} \quad \frac{t^9}{t^3} = t^{9-3} = t^6
\]

So, \(\frac{18t^9}{-6t^3} = -3t^6\).

### Step 2: Combine the results.

\[
2t^4 - 3t^6
\]

Now let's check the answer options:

1. \(-3t^{12} - 12t^7\) - This is not correct.
2. \(-2t^3 - 3t\) - This is not correct.
3. \(-3t^6(4t - 1)\) - Let's expand this:

\[
-3t^6(4t - 1) = -12t^7 + 3t^6
\]

This does not match \(2t^4 - 3t^6\).

4. \(t^4(2 - 3t^2)\) - Let's expand this:

\[
t^4(2 - 3t^2) = 2t^4 - 3t^6
\]

This matches the simplified expression.

### Correct Answer:
The correct answer is:

\[
\boxed{t^4(2 - 3t^2)}
\]

Would you like further details on any of these steps?

### Related Questions:
1. How do you factor higher-degree polynomials like \(t^4(2 - 3t^2)\)?
2. What are the rules for simplifying exponents during division?
3. How do you apply the distributive property when expanding terms?
4. Can you explain more about combining like terms in polynomial expressions?
5. What are some common mistakes to avoid when simplifying expressions with negative signs?

### Tip:
Always watch out for signs and double-check your exponent rules when simplifying expressions involving variables.

-12t^7 18t^9 over - 6t^3 simplified is? -3t^12 - 12t^7, - 2t^3 - 3t, - 3t^6(4t-1), t^4(2-3t^2). Which answer is correct?

Simplify the expression (-12t^7 + 18t^9) / -6t^3 step-by-step and find the correct answer from multiple choices. Explore exponent rules and the distributive property. Suitable for high school students in Grades 9-12.

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