Question

one root of $f(x)=x^3 + 10x^2 - 25x - 250$ is $x = -10$. what are all the roots of the function? use the remainder theorem.
\\(\bigcirc\\) $x = -25$ or $x = 10$
\\(\bigcirc\\) $x = -25$, $x = 1$, or $x = 10$
\\(\bigcirc\\) $x = -10$ or $x = 5$
\\(\bigcirc\\) $x = -10$, $x = -5$, or $x = 5$

Explanation:

Step1: Factor the polynomial using the known root

Since \( x = -10 \) is a root, \( (x + 10) \) is a factor. We perform polynomial division or use synthetic division to factor \( f(x) = x^3 + 10x^2 - 25x - 250 \). Using synthetic division with root \( -10 \):

$$
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So, \( f(x) = (x + 10)(x^2 - 25) \).

Step2: Factor the quadratic

We recognize \( x^2 - 25 \) as a difference of squares, which factors as \( (x - 5)(x + 5) \). So, \( f(x) = (x + 10)(x - 5)(x + 5) \).

Step3: Find the roots

To find the roots, set each factor equal to zero:

• \( x + 10 = 0 \) gives \( x = -10 \)
• \( x - 5 = 0 \) gives \( x = 5 \)
• \( x + 5 = 0 \) gives \( x = -5 \)

Thus, the roots are \( x = -10 \), \( x = -5 \), and \( x = 5 \).

Answer:

D. \( x = -10 \), \( x = -5 \), or \( x = 5 \)