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

To find the square root of the expression \((3t^2 - 2t + 1)(t^2 - 1)\), we can proceed step-by-step. The goal is to simplify this expression under the square root if possible.

### Step 1: Simplify the Expression Under the Square Root

First, let's write the expression:

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

We want to check if we can simplify the product \((3t^2 - 2t + 1)(t^2 - 1)\) under the square root.

### Step 2: Expand the Product

Let's expand the product to see if the expression can be simplified:

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

Expand by multiplying each term in the first polynomial by each term in the second polynomial:

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

Simplify each term:

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

Combine like terms:

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

### Step 3: Look for a Perfect Square

To derive the square root, we need to check if the polynomial \(3t^4 - 2t^3 - 2t^2 + 2t - 1\) is a perfect square. 

Let's assume that there is a polynomial \(P(t) = at^2 + bt + c\) such that:

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

Expanding \(P(t)^2\), we have:

\[
(at^2 + bt + c)^2 = a^2t^4 + 2abt^3 + (2ac + b^2)t^2 + 2bct + c^2
\]

Comparing this with \(3t^4 - 2t^3 - 2t^2 + 2t - 1\), we equate the coefficients:

- \(a^2 = 3\)
- \(2ab = -2\)
- \(2ac + b^2 = -2\)
- \(2bc = 2\)
- \(c^2 = -1\)

Notice here, \(c^2 = -1\) is impossible for real numbers \(c\), meaning the expression is not a perfect square.

### Step 4: Conclusion

Since the polynomial under the square root is not a perfect square, we conclude that the square root of the expression remains:

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

### Summary

The square root cannot be simplified further in terms of elementary functions, so the final expression is:

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

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 determine if a polynomial is a perfect square?
2. What techniques are useful for simplifying expressions under a square root?
3. How does one factor more complex polynomials?
4. Can complex numbers help simplify expressions involving square roots?
5. What methods are available for approximating the square root of polynomials?

**Tip:** When simplifying expressions under a square root, always check if the expression is a perfect square. If not, consider if further algebraic manipulation is possible or if numerical methods might be needed.

Discover how to find the square root of the expression (3t²-2t+1)(t²-1) with a step-by-step solution. This problem involves simplifying polynomials under a square root, suitable for advanced high school students.

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