8. [-/1 Points] DETAILS MY NOTES TGEIALG6 6.3.086. Factor. If an expression is prime, so indicate. (If the expression is not factorable using integers, enter PRIME. \( -14 t^{2}+1+5 t \) SUBMIT ANSWER

To factor the expression \(-14t^2 + 5t + 1\), we can rearrange terms to get \(-14t^2 + 5t + 1 = -14t^2 + 5t + 1\). By using factoring techniques such as grouping or the quadratic formula, you can analyze the discriminant to determine if the expression can be factored using integers. Here, however, the discriminant \((b^2 - 4ac = 5^2 - 4(-14)(1))\) evaluates to a positive value, indicating that it may factor, but on breaking it down, we find it does not yield integer factors. Thus, we conclude that this expression is PRIME. In the world of mathematics, a prime polynomial means it cannot be factored into simpler polynomials with integer coefficients, similar to prime numbers in arithmetic that cannot be divided evenly by any other numbers except for one and themselves. So next time you're faced with a polynomial, check those roots and discriminants closely for possible factorizations!