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$119$ is a root of $f(x) = x^2 + 14,161$ . 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.

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Complex conjugate theorem

Full solution

Q.

119

is a root of

f(x) = x^2 + 14,161

. 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.

Write as $x - 119$ : Since $119$ is a root, we can write it as $x - 119 = 0$ . Now we need to find the other root by factoring the polynomial. $\newline$ $f(x) = x^2 + 14,161$ can be rewritten as $f(x) = (x - 119)(x - a)$ where $a$ is the other root we need to find.
Find other root: To find the value of $a$ , we expand the factored form and compare coefficients. $(x - 119)(x - a) = x^2 - (119 + a)x + 119a$ The constant term in the polynomial $f(x)$ is $14,161$ , so we set $119a$ equal to $14,161$ .
Expand and compare coefficients: Solving for 'a', we divide $14,161$ by $119$ . $\newline$ $a = \frac{14,161}{119}$ $\newline$ $a = 119$

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