Sure, let's find the zeros of the polynomial [tex]\( P(x) = x^3 - 4x^2 - 19x - 14 \)[/tex] and write it in its factored form step by step.
### Step 1: Finding the Zeros
1. Understanding the Polynomial:
The polynomial given is [tex]\( P(x) = x^3 - 4x^2 - 19x - 14 \)[/tex].
2. Finding Rational Zeros:
To find the rational zeros, we can use the Rational Root Theorem, which suggests that any rational root of the polynomial is a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is -14 and the leading coefficient is 1. Possible rational roots are:
[tex]\[
\pm 1, \pm 2, \pm 7, \pm 14
\][/tex]
3. Testing Possible Zeros:
We test these possible roots by substituting them into the polynomial to see if [tex]\( P(x) = 0 \)[/tex].
- [tex]\( P(-2) = (-2)^3 - 4(-2)^2 - 19(-2) - 14 = -8 - 16 + 38 - 14 = 0 \)[/tex]
So, [tex]\( -2 \)[/tex] is a zero.
- [tex]\( P(-1) = (-1)^3 - 4(-1)^2 - 19(-1) - 14 = -1 - 4 + 19 - 14 = 0 \)[/tex]
So, [tex]\( -1 \)[/tex] is a zero.
- [tex]\( P(7) = (7)^3 - 4(7)^2 - 19(7) - 14 = 343 - 196 - 133 - 14 = 0 \)[/tex]
So, [tex]\( 7 \)[/tex] is a zero.
Since [tex]\( -2 \)[/tex], [tex]\( -1 \)[/tex], and [tex]\( 7 \)[/tex] are all zeros, we have found all the zeros of the polynomial.
### Step 2: Writing the Polynomial in Factored Form
1. Using the Zeros:
Given the zeros [tex]\( -2 \)[/tex], [tex]\( -1 \)[/tex], and [tex]\( 7 \)[/tex], the polynomial can be factored as follows:
[tex]\[
P(x) = (x - (-2))(x - (-1))(x - 7) = (x + 2)(x + 1)(x - 7)
\][/tex]
### Conclusion
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]
