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$54$ is a root of $f(x) = x^2 + 2,916$ . Find the other roots of $f(x)$ . $\newline$ Write your answer as a list of simplified values separated by commas, if there is more than one value. $\newline$ ______ $\newline$

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Complex conjugate theorem

Full solution

Q.

54

is a root of

f(x) = x^2 + 2,916

. Find the other roots of

f(x)

.

\newline

Write your answer as a list of simplified values separated by commas, if there is more than one value.

\newline

______

\newline

Identify Root Factor: Since $54$ is a root, we can write one factor of $f(x)$ as $(x - 54)$ .
Set Up Equation: To find the other root, we need to set up the equation $x^2 + 2,916 = (x - 54)(x - r)$ , where $r$ is the other root.
Expand Right Side: Expanding the right side, we get $x^2 - 54x - rx + 54r$ .
Ensure Linear Term is $0$ : Since the polynomial is $x^2 + 2,916$ , the linear term (the $x$ term) must be $0$ , so $-54 - r = 0$ .
Solve for Other Root: Solving for $r$ , we get $r = -54$ .
Final Roots: Now we have both roots: $54$ and $-54$ .

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