Question

Find all the zeros of the polynomial x³ + 3x² − 2x − 6, if two of its zeros are `-sqrt2` and `sqrt2`

Answer 1

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

Since, `sqrt2` and `-sqrt2` are zeros of f(x).

Therefore

`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`

= x² - 2

x² - 2 is a factor of f(x). Now, we divide x³ + 3x² − 2x − 6 by g(x) = x² - 2 to find the zero of f(x).

By using division algorithm we have

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

x³ + 3x² − 2x − 6 = (x² - 2)(x + 3) - 0

x³ + 3x² − 2x − 6 `=(x+sqrt2)(x-sqrt2)(x+3)`

Hence, the zeros of the given polynomials are `-sqrt2`, `+sqrt2` and  -3.