Final answer:
To factor the trinomial 3x²+16x+21, we use the ac method. After finding the numbers whose product is 63 and sum is 16, we split the middle term and factor by grouping to get the final factored form (3x+7)(x+3).
Step-by-step explanation:
To factor the trinomial 3x²+16x+21, we need to find two binomials that multiply together to give us the trinomial. First, we check if the trinomial can be factored using the ac method. To do this, we multiply the coefficient of x² (3) by the constant term (21) which gives us 63. Now, we need to find two numbers whose product is 63 and whose sum is the coefficient of x (16). After some trial and error, we find that 7 and 9 are the numbers we need.
Next, we split the middle term by replacing the coefficient of x (16x) with the sum of our two numbers (7x+9x). This gives us the expression 3x²+7x+9x+21.
Now, we can factor by grouping. We group the first two terms together and the last two terms together. From the first group, we can factor out the greatest common factor, which is x. From the second group, we can factor out the greatest common factor, which is 7. This gives us x(3x+7)+7(3x+7).
Finally, we have two binomials, (3x+7) and (7), that are common to both terms. We can factor these out to get the final factored form:
3x²+16x+21 = (3x+7)(x+3).