To determine the third root of the polynomial equation [tex]\(x^3 + x^2 + 9x + 9 = 0\)[/tex], given that the roots [tex]\( \pm 3i \)[/tex] are known, we can follow a systematic approach using the properties of polynomials and solving for the unknown root.
### Step-by-Step Solution:
1. Identify the Polynomial and Known Roots:
The polynomial is:
[tex]\[
P(x) = x^3 + x^2 + 9x + 9
\][/tex]
The known roots are:
[tex]\[
x_1 = 3i \quad \text{and} \quad x_2 = -3i
\][/tex]
2. Sum of the Roots of the Polynomial:
By Vieta's formulas, the sum of the roots (considering the polynomial of degree 3) is given by the coefficient of [tex]\(x^2\)[/tex] with a sign change:
[tex]\[
x_1 + x_2 + x_3 = -\left(\text{coefficient of } x^2 / \text{leading coefficient}\right)
= -\left(1 / 1\right) = -1
\][/tex]
3. Calculate the Known Sum of Roots:
The sum of the known roots [tex]\(3i\)[/tex] and [tex]\(-3i\)[/tex] is:
[tex]\[
3i + (-3i) = 0
\][/tex]
4. Determine the Third Root:
Let the third root be [tex]\(x_3\)[/tex]. According to Vieta's formulas:
[tex]\[
3i + (-3i) + x_3 = -1
\][/tex]
Simplifying, we find:
[tex]\[
0 + x_3 = -1 \implies x_3 = -1
\][/tex]
Therefore, the other (third) root of the polynomial [tex]\(P(x) = x^3 + x^2 + 9x + 9\)[/tex] is:
[tex]\[
\boxed{-1}
\][/tex]
By substituting this back into the polynomial to verify, we see:
[tex]\[
P(-1) = (-1)^3 + (-1)^2 + 9(-1) + 9 = -1 + 1 - 9 + 9 = 0
\][/tex]
Thus, the calculation confirms that [tex]\(-1\)[/tex] is indeed a root of the polynomial, ensuring our solution is correct.
