Question

Obtain all zeros of the polynomial f(x) = x⁴ − 3x³ − x² + 9x − 6, if two of its zeros are ${\displaystyle -\sqrt{3}}$ and ${\displaystyle \sqrt{3}}$

Answer 1

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

since ${\displaystyle -\sqrt{3}}$ and ${\displaystyle \sqrt{3}}$ are zeros of f(x).

Therefore

${\displaystyle (x+\sqrt{3})(x-\sqrt{3})=x^2+\sqrt{3}x-\sqrt{3}x-3}$

= x² - 3

x² - 3 is a factor of f(x). Now , we divide f(x) = x⁴ − 3x³ − x² + 9x − 6 by g(x) = x² - 3 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 − 3x³ − x² + 9x − 6 = (x² - 3)(x² - 3x + 2) + 0

x⁴ − 3x³ − x² + 9x − 6 = (x² - 3)(x² - 2x + 1x + 2)

x⁴ − 3x³ − x² + 9x − 6 = (x² - 3)[x(x - 2) - 1(x - 2)]

x⁴ − 3x³ − x² + 9x − 6 = (x² - 3)[(x - 1)(x - 2)]

x⁴ − 3x³ − x² + 9x − 6 ${\displaystyle =(x-\sqrt{3})(x+\sqrt{3})(x-1)(x-2)}$

Hence, the zeros of the given polynomials are ${\displaystyle -\sqrt{3}}$, ${\displaystyle \sqrt{3}}$, +1 and +2.