What are the characteristics and key properties of the polynomial y² + 3y + 12?

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The polynomial in question is y² + 3y + 12. Let’s break down its key characteristics:

  • Degree: The degree of this polynomial is 2 because the highest exponent of the variable y is 2.
  • Leading Coefficient: The leading coefficient (the coefficient of the term with the highest degree) is 1, which means that the polynomial is a quadratic function that opens upwards.
  • Roots: To find the roots of the polynomial, we can use the quadratic formula: y = (-b ± √(b² – 4ac)) / (2a), where a = 1, b = 3, and c = 12. Plugging in these values gives:
y = (-3 ± √(3² - 4 × 1 × 12)) / (2 × 1)
   = (-3 ± √(9 - 48)) / 2
   = (-3 ± √(-39)) / 2

Since the discriminant (the part under the square root) is negative (-39), this polynomial has no real roots. Instead, it has two complex roots, which can be expressed as:

y = (-3 ± i√39) / 2
  • Shape of the Graph: The graph of the polynomial is a parabola that opens upwards, indicating that it has a minimum point. Since there are no real roots, the parabola does not intersect the x-axis.
  • Axis of Symmetry: The axis of symmetry for this quadratic function can be found using the formula x = -b / (2a). For our polynomial, it is:
x = -3 / (2 × 1) = -3/2

This axis of symmetry indicates the vertical line around which the parabola is symmetric.

  • Y-Intercept: To find the y-intercept of the polynomial, substitute y = 0 into the polynomial:
y² + 3y + 12 → 0² + 3(0) + 12 = 12

Thus, the y-intercept is at the point (0, 12).

In summary:

  • The polynomial has a degree of 2.
  • It has one leading coefficient of 1, indicating an upwards-opening parabola.
  • It has no real roots and two complex roots.
  • The axis of symmetry is at y = -3/2.
  • The y-intercept is at (0, 12).

Understanding these properties provides a solid foundation for analyzing the polynomial and its graph.

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