Question

One root of f (x) = x cubed + 10 x squared minus 25 x minus 250 is x = –10. What are all the roots of the function? Use the Remainder Theorem.
x = –25 or x = 10
x = –25, x = 1, or x = 10
x = –10 or x = 5
x = –10, x = –5, or x = 5

Answer 1

If x = -10 is a root, divide f(x) by (x + 10). Synthetic division with -10:

Coefficients: 1, 10, -25, -250
Bring down 1; multiply -10 → -10; add to 10 → 0; multiply -10 → 0; add to -25 → -25; multiply -10 → 250; add to -250 → 0.

Quotient is x^2 - 25, so
f(x) = (x + 10)(x^2 - 25) = (x + 10)(x - 5)(x + 5).

Roots: x = -10, -5, 5.