Questions: 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 x= Sketch the graph.

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

x=

Sketch the graph.

Transcript text: 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.) \[ P(x)=x^{3}-2 x^{2}-15 x \] $x=$ $\square$ Sketch the graph.

Solution

Solution Steps

Step 1: Factor the Polynomial

The given polynomial is

\[ P(x) = x^3 - 2x^2 - 15x \]

First, factor out the greatest common factor, which is $x$:

\[ P(x) = x(x^2 - 2x - 15) \]

Next, factor the quadratic expression $x^2 - 2x - 15$. We need two numbers that multiply to $-15$ and add to $-2$. These numbers are $-5$ and $3$. Therefore, the quadratic can be factored as:

\[ x^2 - 2x - 15 = (x - 5)(x + 3) \]

Thus, the factored form of the polynomial is:

\[ P(x) = x(x - 5)(x + 3) \]

Step 2: Find the Zeros of the Polynomial

To find the zeros, set each factor equal to zero:

$x = 0$
$x - 5 = 0 \Rightarrow x = 5$
$x + 3 = 0 \Rightarrow x = -3$

The zeros of the polynomial are $x = 0$, $x = 5$, and $x = -3$.

Final Answer

The zeros of the polynomial are $x = 0, 5, -3$.

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