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$93\sqrt{5}$ is a root of $f(x) = x^2 - 43,245$ . 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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Conjugate root theorems

Full solution

Q.

93\sqrt{5}

is a root of

f(x) = x^2 - 43,245

. 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.

Identify Conjugate Pairs: Since $93\sqrt{5}$ is a root, the other root will also be a radical because polynomials have roots that come in conjugate pairs when they involve radicals.
Determine Other Root: The other root will be $-93\sqrt{5}$ because the sum of the roots is the negation of the coefficient of the $x$ term in the polynomial, which is $0$ since there is no $x$ term.
Calculate Constant Term: To check, we can multiply the roots together to get the constant term of the polynomial: $93\sqrt{5}$ ( $-93$ \sqrt{ $5$ }) = - $93^2 \times 5$ .
Calculate Constant Term: To check, we can multiply the roots together to get the constant term of the polynomial: $93\sqrt{5}$ ( $-93$ \sqrt{ $5$ }) = - $93^2 \times 5$ .Calculate the multiplication: $93^2 \times 5 = 8649 \times 5 = 43245$ .

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