The polynomial you provided is 2x² + 3x + 5. To find its factors, we need to analyze its structure and determine if it can be factored into simpler expressions.
Firstly, we look for two numbers that multiply to the product of the coefficient of x² (which is 2) and the constant term (which is 5). The product is 2 * 5 = 10. Next, we also need these two numbers to add up to the coefficient of the x term, which is 3.
Unfortunately, no such pair of numbers exists that fulfills both conditions. Therefore, this polynomial does not factor neatly into rational numbers.
However, we can still analyze its nature further. We can complete the square or apply the quadratic formula to find its roots:
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)In this case, a = 2, b = 3, and c = 5. Plugging these values into the formula gives:
x = ( -3 ± √(3² - 4 * 2 * 5) ) / (2 * 2)This simplifies to:
x = ( -3 ± √(9 - 40) ) / 4Since the discriminant (9 – 40) is negative (-31), it indicates that the polynomial has no real roots, thus confirming that it cannot be factored over the real numbers.
If you’re looking for factors in terms of complex numbers, the polynomial can be expressed with complex roots, but in terms of simple factors with real coefficients, it is irreducible. Therefore, the polynomial 2x² + 3x + 5 is not factorable into simpler polynomial factors.