Final answer:
To factor the trinomial 6x²+13x+2, we find the right combination of factors of 12 that both multiply to 12 and add to 13, which are 1 and 12. By rewriting the middle term and grouping, we finally factor the trinomial as (6x + 1)(x + 2).
Step-by-step explanation:
To factor the trinomial 6x²+13x+2 over the integers, we need to find two numbers that multiply to give the product of the coefficient of x² and the constant term (6*2=12), and at the same time add up to the coefficient of the x term, which is 13.
Let's find the numbers:
1. 1 and 12 (1*12=12, 1+12=13)
2. 2 and 6 (2*6=12, 2+6=8)
3. 3 and 4 (3*4=12, 3+4=7)
None of these pairs add up to 13, so we need to try different factors of 12.
After trying different combinations, we find that the correct numbers are 1 and 12. We can now rewrite the middle term (13x) as the sum of two terms whose coefficients are the numbers we found:
6x² + 1x + 12x + 2
Now, group the terms and factor by grouping:
(6x² + 1x) + (12x + 2)
= x(6x + 1) + 2(6x + 1)
Now factor out the common binomial factor:
= (6x + 1)(x + 2)
Thus, the trinomial factors to (6x + 1)(x + 2).