To solve the given problem [tex]\( P(x) = x^3 - 4x^2 - 19x - 14 \)[/tex], we need to find the zeros of the polynomial and write it in factored form. Here's a step-by-step approach to achieve this:
### Step 1: Find the Zeros of the Polynomial
The zeros of a polynomial are the values of [tex]\( x \)[/tex] that make [tex]\( P(x) = 0 \)[/tex]. We will solve the equation [tex]\( x^3 - 4x^2 - 19x - 14 = 0 \)[/tex] for [tex]\( x \)[/tex]. The solutions are:
[tex]\[ x = -2, -1, 7 \][/tex]
Hence, the zeros of the polynomial [tex]\( P(x) \)[/tex] are [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 7 \)[/tex].
### Step 2: Write the Polynomial in Factored Form
Given the zeros [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 7 \)[/tex], the polynomial can be expressed in its factored form by creating factors in the form [tex]\( (x - \text{zero}) \)[/tex]. Consequently, the polynomial [tex]\( P(x) \)[/tex] can be written as:
[tex]\[
P(x) = (x - (-2))(x - (-1))(x - 7)
\][/tex]
Simplifying the factors, we have:
[tex]\[
P(x) = (x + 2)(x + 1)(x - 7)
\][/tex]
Thus, the polynomial [tex]\( P(x) \)[/tex] in its factored form is:
[tex]\[
P(x) = (x + 2)(x + 1)(x - 7)
\][/tex]
### Summary
- The zeros of the polynomial [tex]\( P(x) = x^3 - 4x^2 - 19x - 14 \)[/tex] are [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 7 \)[/tex].
- The polynomial in its factored form is [tex]\( P(x) = (x + 2)(x + 1)(x - 7) \)[/tex].
This detailed procedure provides the requested solutions through clear and logical steps.
