Factor the polynomial and use the factored form to find the zeros, (Enter your answers as a comma-separated list, Enter all answers using the appropriate multiplicities.) \[ P(x)=x^{3}-2 x^{2}-15 x \] Sketch the graph.

To factor the polynomial \( P(x) = x^3 - 2x^2 - 15x \), we can first factor out the greatest common factor, which is \( x \): \[ P(x) = x(x^2 - 2x - 15) \] Next, we will factor the quadratic \( x^2 - 2x - 15 \). We need two numbers that multiply to \(-15\) and add to \(-2\). Those numbers are \(-5\) and \(3\). Therefore, we can factor it as follows: \[ P(x) = x(x - 5)(x + 3) \] Now, we can find the zeros by setting each factor equal to zero: 1. \( x = 0 \) 2. \( x - 5 = 0 \) → \( x = 5 \) 3. \( x + 3 = 0 \) → \( x = -3 \) Thus, the zeros of the polynomial are \( 0, 5, -3 \). Since each factor is linear, each zero has multiplicity 1. Sketching the graph: - The polynomial \( P(x) \) is a cubic function, which means it will have one end going to positive infinity and the other going to negative infinity. - The zeros \( x = 0 \), \( x = 5 \), and \( x = -3 \) will be the x-intercepts. - The function will cross the x-axis at these points, with the cubic shape curving up and down around these intercepts. So the rough sketch should show a curve starting in the bottom left, passing through \( (-3, 0) \), peaking, crossing through \( (0, 0) \), dipping back down, and then crossing through \( (5, 0) \) while heading up towards the top right.