$-70i$ 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. $\newline$ ______

Full solution

-70i

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

______

Given Polynomial: The given polynomial is $f(x) = x^2 + 4,900$ . We know that $-70i$ is one of the roots. Since the coefficients of the polynomial are real numbers, the complex roots of polynomials with real coefficients come in conjugate pairs. Therefore, if $-70i$ is a root, then its conjugate, $70i$ , must also be a root.
Conjugate Pairs: To find the other roots, we can use the fact that the product of the roots is equal to the constant term of the polynomial divided by the leading coefficient. For the polynomial $f(x) = x^2 + 4,900$ , the constant term is $4,900$ and the leading coefficient is $1$ . The product of the roots is $4,900$ .
Product of Roots: We can express the product of the roots as $(-70i) \times (70i) = 4,900$ . Simplifying the left side gives us $(-70i) \times (70i) = -4900i^2$ . Since $i^2 = -1$ , this simplifies to $-4900 \times -1 = 4,900$ , which matches the constant term of the polynomial.
Final Roots: Since we have found that the product of the roots is indeed $4,900$ and we have both roots as $-70i$ and $70i$ , we have found all the roots of the polynomial $f(x) = x^2 + 4,900$ .