Problem 27 - College Algebra Essentials

Question

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-11 x+4$$

Step-by-Step Solution

Answer

The trinomial \(6x^{2}-11x+4\) is prime.

Step 1: Identify the Coefficients and Constant

Identify the coefficients of the \(x^{2}\) term and the \(x\) term, and the constant. Here, the coefficient of the \(x^{2}\) term is 6, the coefficient of the \(x\) term is -11, and the constant is 4.

Step 2: Find Two Numbers

The task is to find two numbers that multiply to give the product of the coefficient of the \(x^{2}\) term and the constant term (6 * 4 = 24), and add up to give the coefficient of the \(x\) term, which is -11. After checking the pairs of factors of 24 ((1, 24), (2, 12), (3, 8), (4, 6)), you'll find that none of them add up to -11. Thus, there are no such two numbers.

Step 3: Conclusion

Since the task of finding two numbers that multiply to 24 and add to -11 is unsuccessful, it can be concluded that this trinomial cannot be factored further using integers. Therefore, the trinomial \(6x^{2}-11x+4\) is prime.