Question

Obtain all other zeroes of the polynomial x4 – 3√2x3 – 3x2 + 3√2x – 4, if two of its zeroes are √2 and 2√2.

Answer 1

Step-by-step explanation:

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.