Factor the trinomial: 2x^2 + 7x + 3
To factor the trinomial 2x^2 + 7x + 3, you need to find two binomials that, when multiplied together, will give you the original trinomial.
First, look at the coefficient in front of the x^2 term, which is 2. We need to find two numbers that multiply together to give 2, the coefficient of x^2. The possible combinations are 1 and 2, or -1 and -2.
Next, look at the constant term, which is 3. We need to find two numbers that add up to 7, the coefficient of the x term, and multiply together to give 3. The possible combinations are 1 and 3, or -1 and -3.
Now, you can try different combinations of these numbers to see which combination works.
Option 1: (2x + 1)(x + 3)
If you multiply these two binomials together, you get:
2x * x = 2x^2
2x * 3 = 6x
1 * x = x
1 * 3 = 3
Adding like terms, you get 2x^2 + 7x + 3. This combination works, so the trinomial 2x^2 + 7x + 3 can be factored as (2x + 1)(x + 3).
Option 2: (-2x - 1)(-x - 3)
If you multiply these two binomials together, you get:
-2x * -x = 2x^2
-2x * -3 = 6x
-1 * -x = x
-1 * -3 = 3
Adding like terms, you get 2x^2 + 7x + 3. This combination also works, so the trinomial 2x^2 + 7x + 3 can be factored as (-2x - 1)(-x - 3).
Therefore, the factored form of the trinomial 2x^2 + 7x + 3 is (2x + 1)(x + 3) or (-2x - 1)(-x - 3).