Find all zeros of the polynomial 2x^{4} + 7x^{3} − 19x^{2} − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.

#### Solution

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`

= x^{2} - 2

x^{2} - 2 is a factor of f(x). Now, we divide 2x^{4} + 7x^{3} − 19x^{2} − 14x + 30 by g(x) = x^{2} - 2 to find the zero of f(x).

By using division algorithm we have

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

2x^{4} + 7x^{3} − 19x^{2} − 14x + 30 = (x^{2} - 2)(2x^{2} + 7x - 15) + 0

2x^{4} + 7x^{3} − 19x^{2} − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x^2+10x-3x-15)`

2x^{4} + 7x^{3} − 19x^{2} − 14x + 30 `=(x+sqrt2)(x-sqrt2)[2x(x+5)-3(x+5)]`

2x^{4} + 7x^{3} − 19x^{2} − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x-3)(x+5)`

Hence, the zeros of the given polynomial are `-sqrt2`, `+sqrt2`, `(+3)/2`, -5.