Question

Find all zeros of the polynomial f(x) = 2x⁴ − 2x³ − 7x² + 3x + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`

Answer 1

Since `-sqrt(3/2)` and `sqrt(3/2)` are two zeros of f(x) Therefore,

`=(x-sqrt(3/2))(x+sqrt(3/2))`

`=(x^2-3/2)`

`=1/2(2x^2-3)` is a factor of f(x).

Also 2x² - 3 is a factor of f(x).

Let us now divide f(x) by 2x² - 3. we have

By using that division algorithm we have,

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

2x⁴ − 2x³ − 7x² + 3x + 6 = (2x² - 3)(x² - x - 2) + 0

2x⁴ − 2x³ − 7x² + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)(x^2+1x-2x-2)`

2x⁴ − 2x³ − 7x² + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)[x(x+1)-2(x+1)]`

2x⁴ − 2x³ − 7x² + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)(x-2)(x+1)`

Hence, The zeros of f(x) are `-sqrt(3/2)`, `sqrt(3/2)`, 2 , -1.