Questions: Sketch the graph of the polynomial function. Do not use a graphing utility. f(x) = x^3 + 15x + 80 (in factored form, f(x) = -(x + 5)(x + 4)(x - 5))

Solution Steps

The given polynomial function is \( P(x) = x^3 - 3x^2 - 9x + 27 \).

Factor the polynomial to find its roots. The polynomial can be factored as: \[ P(x) = (x - 3)(x + 3)(x - 3) \] or \[ P(x) = (x - 3)^2(x + 3) \]

The roots of the polynomial are \( x = 3 \) (with multiplicity 2) and \( x = -3 \) (with multiplicity 1).

• At \( x = 3 \), since the root has even multiplicity, the graph touches the x-axis and turns around.
• At \( x = -3 \), since the root has odd multiplicity, the graph crosses the x-axis.

Since the leading term of the polynomial is \( x^3 \), the end behavior is:

• As \( x \to \infty \), \( P(x) \to \infty \)
• As \( x \to -\infty \), \( P(x) \to -\infty \)

Based on the roots, their multiplicities, and the end behavior, the correct graph should:

• Cross the x-axis at \( x = -3 \)
• Touch and turn around at \( x = 3 \)
• Have the end behavior of going to \( \infty \) as \( x \to \infty \) and going to \( -\infty \) as \( x \to -\infty \)

Final Answer