Derive the square root of (3t²-2t+1)(t²-1)

To find the square root of the expression \((3t^2 - 2t + 1)(t^2 - 1)\), we'll first simplify the expression inside the square root and then find its square root.

### Step 1: Expand the Expression Inside the Square Root

First, expand the expression \((3t^2 - 2t + 1)(t^2 - 1)\).

\[
(3t^2 - 2t + 1)(t^2 - 1) = 3t^2(t^2 - 1) - 2t(t^2 - 1) + 1(t^2 - 1)
\]

Distribute each term:

\[
= 3t^4 - 3t^2 - 2t^3 + 2t + t^2 - 1
\]

Combine like terms:

\[
= 3t^4 - 2t^3 - 2t^2 + 2t - 1
\]

### Step 2: Apply the Square Root

Now, we want to find the square root of this expression:

\[
\sqrt{3t^4 - 2t^3 - 2t^2 + 2t - 1}
\]

### Step 3: Simplify the Expression (if possible)

Finding a simplified form for the square root of a polynomial can be challenging. Let's check if the polynomial can be factored into a perfect square. 

To express \(\sqrt{3t^4 - 2t^3 - 2t^2 + 2t - 1}\) in a simpler form, we need to find if there exists a polynomial \(P(t)\) such that:

\[
P(t)^2 = 3t^4 - 2t^3 - 2t^2 + 2t - 1
\]

However, upon examination, the polynomial \(3t^4 - 2t^3 - 2t^2 + 2t - 1\) does not appear to be a perfect square.

### Conclusion

The expression \(\sqrt{(3t^2 - 2t + 1)(t^2 - 1)}\) simplifies to:

\[
\sqrt{3t^4 - 2t^3 - 2t^2 + 2t - 1}
\]

Since the polynomial under the square root is not a perfect square, this is the simplified form. There is no further simplification unless specific values of \(t\) are provided.

Would you like more details on this solution, or do you have any questions? Here are 5 questions that might help you explore further:

1. How do you factor polynomials into simpler forms?
2. What methods can be used to check if a polynomial is a perfect square?
3. How can you use derivatives to analyze the behavior of a polynomial function?
4. What are some practical applications of polynomial expressions in real-world problems?
5. How do you graph a polynomial function to visualize its roots and behavior?

**Tip:** When working with square roots of expressions, always check if the expression can be simplified by factoring or completing the square.

Discover how to find the square root of the expression (3t² - 2t + 1)(t² - 1) with detailed steps and explanations. Explore polynomial operations and square root simplifications in this advanced math problem. Suitable for Grades 10-12.

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