Given the function \( P(x)=x^{3}+x^{2}-6 x \) The \( y \)-intercept is The \( x \)-intercepts is/are When \( x \rightarrow \infty, y \rightarrow ? \) ? When \( x \rightarrow-\infty, y \rightarrow ? \)

To find the \( y \)-intercept of the function \( P(x) = x^3 + x^2 - 6x \), substitute \( x = 0 \): \[ P(0) = 0^3 + 0^2 - 6(0) = 0 \] So, the \( y \)-intercept is \( (0, 0) \). For the \( x \)-intercepts, set \( P(x) = 0 \): \[ x^3 + x^2 - 6x = 0 \] Factoring out \( x \): \[ x(x^2 + x - 6) = 0 \] This gives \( x = 0 \), and for \( x^2 + x - 6 = 0 \), apply the quadratic formula: \[ x = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm 5}{2} \] Thus, \( x = 2 \) and \( x = -3 \). The \( x \)-intercepts are \( (0, 0) \), \( (2, 0) \), and \( (-3, 0) \). As \( x \to \infty \), the dominant term \( x^3 \) dictates that \( y \to \infty \). Conversely, as \( x \to -\infty \), again the dominant term leads to \( y \to -\infty \). So, \( y \to \infty \) as \( x \to \infty \), and \( y \to -\infty \) as \( x \to -\infty \). --- The polynomial function we’re dealing with is a classic example of cubic behavior, which means it can have up to three \( x \)-intercepts—how cool is that? Each of those intercepts gives us critical information about the function's graph! Cubic functions also have fascinating properties like the ability to change direction up to two times, allowing for those lovely wiggles and turns on the graph. When tackling polynomial equations, it's easy to overlook factoring. A quick factorization can save you time and headaches. A common mistake is trying to apply the quadratic formula without first simplifying the equation as much as possible. Maintaining awareness of symmetry in polynomial graphs can also provide insight—checking the endpoints helps paint a fuller picture of the function's behavior across its domain!