Factor the trinomial: 3x^2 + 11x + 6
To factor the trinomial 3x^2 + 11x + 6, we look for two binomials in the form (ax + c)(bx + d) that, when multiplied together, give us the original trinomial.
We start by finding the factors of the leading coefficient (3) and the constant term (6). In this case, the factors of 3 are 1 and 3, and the factors of 6 are 1, 2, 3, and 6.
Next, we need to determine which pairs of these factors, when combined, give us the middle term coefficient (11). The possibilities are:
1 * 3 = 3
3 * 1 = 3
1 * 2 = 2
2 * 1 = 2
1 * 6 = 6
6 * 1 = 6
2 * 3 = 6
3 * 2 = 6
To find the correct pair, we look for the pair that sums up to the coefficient of the middle term (11), which in this case is 2 and 3.
The next step is to rewrite the middle term (11x) using these two numbers (2 and 3). We replace the middle term with the sum of these numbers and factor by grouping.
3x^2 + 2x + 3x + 6
Now, we factor by grouping by grouping the terms as follows:
(3x^2 + 2x) + (3x + 6)
Taking out the greatest common factor (GCF) from each group:
x(3x + 2) + 3(3x + 2)
Notice that we now have a common binomial factor of (3x + 2).
Finally, we rewrite the factored form by combining the common factor and the grouped terms:
(3x + 2)(x + 3)
Therefore, the factored form of the trinomial 3x^2 + 11x + 6 is (3x + 2)(x + 3).