One of the numbers In your answer Is Incorrect. $\newline$ Find all other zeros of $\newline$ $P(x)=x^{3}-6x^{2}+18x-40$ , given that $\newline$ $1-3i$ is a zero. $\newline$ (If there is more than one zero, separate them with commas.)

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Q. One of the numbers In your answer Is Incorrect.

\newline

Find all other zeros of

\newline

P(x)=x^{3}-6x^{2}+18x-40

, given that

\newline

1-3i

is a zero.

\newline

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

Identify Complex Conjugate Zeros: Since complex roots of polynomials with real coefficients come in conjugate pairs, if $1 - 3i$ is a zero, then its conjugate, $1 + 3i$ , must also be a zero.
Write Corresponding Factors: We can write the factors corresponding to these zeros as $(x - (1 - 3i))$ and $(x - (1 + 3i))$ .
Multiply Factors for Quadratic Factor: Now we will multiply these factors to find the quadratic factor of the polynomial $P(x)$ . $(x - (1 - 3i))(x - (1 + 3i)) = ((x - 1) + 3i)((x - 1) - 3i)$
Apply Difference of Squares: Using the difference of squares, we get: $\newline$ $((x - 1) + 3i)((x - 1) - 3i) = (x - 1)^2 - (3i)^2$
Simplify Quadratic Factor: Calculating the squares and simplifying: $\newline$ $(x - 1)^2 - (3i)^2 = (x^2 - 2x + 1) - (-9)$ $\newline$ $= x^2 - 2x + 10$
Find Remaining Real Root: Now we have a quadratic factor $x^2 - 2x + 10$ of the polynomial $P(x)$ . Since $P(x)$ is a cubic polynomial, there must be one more real root. We can find this by dividing $P(x)$ by the quadratic factor.
Perform Polynomial Division: Performing the division of $P(x)$ by $x^2 - 2x + 10$ : $\frac{P(x)}{x^2 - 2x + 10} = \frac{x^3 - 6x^2 + 18x - 40}{x^2 - 2x + 10}$
Use Coefficients for Linear Factor: Using polynomial long division or synthetic division, we find the quotient. However, to save time, we can also use the fact that the coefficients of $x^2$ in $P(x)$ and the quadratic factor are the same, so the remaining linear factor must be $x - 4$ .
Determine Last Zero: The remaining linear factor gives us the last zero of $P(x)$ . Setting $x - 4$ equal to zero, we find that $x = 4$ is the last zero.
Combine All Zeros: Combining all the zeros, we have $1 - 3i$ , $1 + 3i$ , and $4$ .