The polynomial y² + 3y + 12 is a quadratic expression. To analyze its properties, we can look at several aspects such as its roots, graph, and whether it factored or not.
First, we can determine if the polynomial has real roots by calculating the discriminant (b² – 4ac). In this polynomial, a = 1, b = 3, and c = 12.
Thus, the discriminant is:
D = 3² – 4 * 1 * 12 = 9 – 48 = -39
Since the discriminant is negative, this indicates that the polynomial does not have any real roots. It only has complex roots.
Graphically, the quadratic opens upwards because its leading coefficient (1) is positive. This means the vertex of the parabola will be its minimum point.
In terms of factoring, this polynomial does not factor neatly into linear factors due to the negative discriminant. Thus, it is expressed in its standard form.
Overall, the polynomial y² + 3y + 12 is always positive and does not intersect the x-axis, indicating that there are no real values for y that satisfy y² + 3y + 12 = 0.