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P(x) = x⁴ - 3 √2 x³ + 3 x² + 3√2 x - 4

given that (x - √2) , (x - 2√2) are factors of P(x) as √2 and 2√2 are two zeroes of P(x) = 0.

(x -√2) (x - 2√2) = x² -3√2 x + 4 is a factor of P(x). let A be a constant. We can write the constant term in the second factor by : -4/4 = -1... dividing the constant terms.

let (x² - 3√2 x + 4) (x² + A x -1 ) = P(x)

Now compare the coefficients of x³ : A - 3√2 = -3√2 => A = 0 coefficient of x : 4A + 3√2 = 3√2 => A = 0

so the other factors are : x² - 1 = 0 so x = 1 and -1 are the other factors.

This method of multiplying the factors and comparing coefficients is simple.

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There is a simple method. I overlooked....

sum of all four roots of P(x) is the - of coefficient of x cube. so it is 3√2. The sum of given two roots is 3 √2. So sum of two more roots is 0.

Now product of four roots is the constant term = -4... product of two given roots is 4. so product of two more roots is -4/4 = -1..... Hence the polynomial is simple : x² - (0) x + (-1) roots of this are +1 and -1.