To factor the expression 9x² + 3x + 2, we need to look for two numbers that multiply to give us the product of the coefficient of x² (which is 9) and the constant term (which is 2). This product is 9 * 2 = 18. We also need these two numbers to add up to the coefficient of x, which is 3.
Upon examining the factors of 18:
- 1 × 18
- 2 × 9
- 3 × 6
- 6 × 3
- 9 × 2
- 18 × 1
We can see that there are no two numbers from this list that add up to 3. This means that the expression 9x² + 3x + 2 cannot be factored over the integers.
However, we can still present it in a factored form by looking for irrational or complex factors, or we can say that it is already in its simplest form for integer coefficients:
- The expression is prime, meaning it cannot be factored further using real numbers.
Thus, the conclusion is that 9x² + 3x + 2 is a prime polynomial with no factors over the integers.