Final answer:
The polynomial 36t² - 24t + 4 is partially factorable with integers by factoring out the greatest common factor, resulting in 4(9t² - 6t + 1). The trinomial 9t² - 6t + 1 cannot be factored further using integers, making it a prime polynomial.
Step-by-step explanation:
The student has asked to factor the polynomial 36t² - 24t + 4. To address this task, one potential method is to look for a common factor that can be factored out. In this case, we can see that each term of the polynomial is divisible by 4. Factoring 4 out gives us:
• 4(9t² - 6t + 1)
Next, we examine the resulting trinomial 9t² - 6t + 1 to see if it can be factored further. This trinomial is a quadratic equation, and we could attempt to factor it into the product of two binomials. However, analyzing the trinomial in question reveals that it does not factor nicely with integers. The quadratic formula or completing the square may be used to find that the trinomial has no real roots, which confirms it cannot be factored. Therefore, the original polynomial 36t² - 24t + 4 is only factorable to 4(9t² - 6t + 1), and the trinomial is prime.