Find all other zeros of $\newline$ $P(x)=x^{3}+5x^{2}+8x+6$ , given that $\newline$ $-1-i$ is a zero $\newline$ (If there is more than one zero, separate them with commas.)

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Q. Find all other zeros of

\newline

P(x)=x^{3}+5x^{2}+8x+6

, given that

\newline

-1-i

is a zero

\newline

(If there is more than one zero, separate them with commas.)

Apply Complex Conjugate Root Theorem: Since the polynomial has real coefficients and $-1-i$ is a zero, its complex conjugate $-1+i$ must also be a zero due to the Complex Conjugate Root Theorem.
Find Remaining Zero: To find the remaining zero, we can divide the polynomial by the product of the factors corresponding to the known zeros $-1-i$ and $-1+i$ . The factors are $(x - (-1 - i))$ and $(x - (-1 + i))$ , which simplify to $(x + 1 + i)$ and $(x + 1 - i)$ .
Multiply Quadratic Factors: We multiply the factors $(x + 1 + i)$ and $(x + 1 - i)$ to get the quadratic factor of $P(x)$ . This multiplication gives us $(x + 1 + i)(x + 1 - i) = x^2 + 2x + (1 - i^2) = x^2 + 2x + 2$ , since $i^2 = -1$ .
Perform Polynomial Division: Now we divide $P(x)$ by the quadratic factor $x^2 + 2x + 2$ using polynomial long division or synthetic division.
Identify Quotient and Remainder: Performing the division, we have: $\newline$ $P(x) = (x^3 + 5x^2 + 8x + 6) \div (x^2 + 2x + 2)$ $\newline$ This division should yield a linear polynomial since we are dividing a cubic polynomial by a quadratic polynomial.
Determine Linear Factor: The division process is as follows: $\newline$ $x^3 + 5x^2 + 8x + 6$ $\newline$ - $(x^3 + 2x^2 + 2x)$ $\newline$ ------------------- $\newline$ $3x^2 + 6x + 6$ $\newline$ - $(3x^2 + 6x + 6)$ $\newline$ ----------------- $\newline$ $0$ $\newline$ The quotient is $x + 3$ , and the remainder is $0$ , which means the division is exact.
Determine Linear Factor: The division process is as follows: $\newline$ $x^3 + 5x^2 + 8x + 6$ $\newline$ - $(x^3 + 2x^2 + 2x)$ $\newline$ ------------------- $\newline$ $3x^2 + 6x + 6$ $\newline$ - $(3x^2 + 6x + 6)$ $\newline$ ----------------- $\newline$ $0$ $\newline$ The quotient is $x + 3$ , and the remainder is $0$ , which means the division is exact.The linear factor $x + 3$ corresponds to the remaining zero of $P(x)$ . Setting $x + 3$ equal to zero gives us the zero $(x^3 + 2x^2 + 2x)$ $0$ .