Question

Find all the zeros of the polynomial x⁴ + x³ − 34x² − 4x + 120, if two of its zeros are 2 and −2.

Answer 1

We know that if x = a is a zero of a polynomial, then x - a is a factor of f(x).

Since, 2 and -2 are zeros of f(x).

Therefore

(x + 2)(x - 2) = x² - 2²

= x² - 4

x² - 4 is a factor of f(x). Now, we divide x⁴ + x³ − 34x² − 4x + 120 by g(x) = x² - 4 to find the other zeros of f(x).

By using that division algorithm we have,

f(x) = g(x) x q(x) - r(x)

x⁴ + x³ − 34x² − 4x + 120 = (x² - 4)(x² + x - 30) - 0

x⁴ + x³ − 34x² − 4x + 120 = (x + 2)(x - 2)(x² + 6x - 5x - 30)

x⁴ + x³ − 34x² − 4x + 120 = (x + 2)(x - 2)(x(x + 6) - 5(x + 6))

x⁴ + x³ − 34x² − 4x + 120 = (x + 2)(x - 2)(x + 6)
(x - 5)

Hence, the zeros of the given polynomial are -2, +2, -6 and 5.