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$70$ is a root of $f(x) = x^2 + 4,900$ . 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.

70

is a root of

f(x) = x^2 + 4,900

. 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: Since $70$ is a root, we can write one factor of $f(x)$ as $(x - 70)$ .
Factor Polynomial: To find the other root, we need to factor the polynomial completely. We know that $f(x) = x^2 + 4,900$ can be written as $(x - 70)(x - a) = 0$ , where $a$ is the other root.
Expand Factors: Expanding $(x - 70)(x - a)$ gives us $x^2 - (70 + a)x + 70a$ . This must be equal to $x^2 + 4,900$ .
Compare Coefficients: Comparing coefficients, we get $70a = 4,900$ . To find $a$ , we divide $4,900$ by $70$ .
Calculate Other Root: $a = \frac{4,900}{70} = 70$ .
Final Roots: So, the other root is also $70$ . The roots of the polynomial are $70$ and $70$ .

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