Question

SH equals to x cube + 7 x square - 2 x minus 14 if two of its zeros are minus root 2 and root 2 then the third zero is ​

Answer 1

We are given the following in the question:

f(x) = x^3 + 3x^2 -5x - 15

x = \sqrt{5}, x = -\sqrt{5} are two roots of the given polynomial.

Since the polynomial have a degree three, there would be three zeros of the given polynomial.

We have to find the third zero of the polynomial.

(x-\sqrt{5}), (x + \sqrt5) would be a factor of the given polynomial.

Thus, we can write:

f(x) = x^3 + 3x^2 -5x - 15 = 0\\\text{To find the third root}\\\\g(x) = \dfrac{x^3 + 3x^2 -5x - 15}{(x+\sqrt5)(x-\sqrt5)}\\\\g(x) = \dfrac{x^3 + 3x^2 -5x - 15}{(x^2-5)}\\g(x) = x+3\\g(x) = 0\\\Rightarrow x = -3\\f(-3) = (-3)^3 + 3(-3)^2 -5(-3) - 15 = 0

Thus, -3 is a root of the given polynomial.