To factor the polynomial expression 3x² + 5x + 1, we need to find two numbers that multiply to the product of the coefficient of x² (which is 3) and the constant term (which is 1), so 3 * 1 = 3. Additionally, we need these two numbers to add up to the coefficient of x, which is 5.
The two numbers that meet these conditions are 3 and 1. Accordingly, we can express the equation as:
3x² + 3x + 2x + 1
Next, we can group the terms:
(3x² + 3x) + (2x + 1)
Now, factor out the common terms within each group:
3x(x + 1) + 1(x + 1)
Notice that both groups contain a common binomial factor of (x + 1). We can factor this out:
(x + 1)(3x + 1)
As a result, the complete factorization of the polynomial expression 3x² + 5x + 1 into its prime factors is:
(x + 1)(3x + 1)
In summary:
- Original expression: 3x² + 5x + 1
- Factored form: (x + 1)(3x + 1)
This shows that the polynomial is factored completely.