To simplify the rational expression \( \frac{t^2}{6t^2 + 9} \), we’ll start by factoring the denominator.
The expression in the denominator, \( 6t^2 + 9 \), can be factored by taking out the greatest common factor:
- Factoring out a 3: \( 6t^2 + 9 = 3(2t^2 + 3) \)
Now the original expression can be rewritten as:
\( \frac{t^2}{3(2t^2 + 3)} \)
Next, let’s discuss any restrictions on the variable \( t \). For a rational expression, we need to ensure that the denominator does not equal zero, as division by zero is undefined.
Setting the denominator equal to zero:
- \( 3(2t^2 + 3) = 0 \)
- Solving for \( t \):\
- \( 2t^2 + 3 = 0 \)
- \( 2t^2 = -3 \)
- Since \( 2t^2 \) is always non-negative for real values of \( t \), this equation does not yield any real solutions.
Thus, the denominator will never be zero for any real number value of \( t \). Therefore, there are no restrictions on the variable \( t \) for this expression.
In summary, the simplified form of the rational expression is:
\( \frac{t^2}{3(2t^2 + 3)} \)
And there are no restrictions on the variable \( t \).