Question

Find all other zeros of the polynomial p(x) = 2x^4-7x^3+19x^2-14x+13 if 2 of its zeros are √2 & -√2

Answer 1

Find all other zeros of the polynomial p(x) = 2x^4-7x^3+19x^2-14x+13 if 2 of its zeros are √2 & -√2
Since √2 and -√2 are zeros of p(x) it means, by factor theorem, that p(x) has factors
(x-√2) and (x+√2). So p(x) is divisible by (x-√2)(x+√2) = x^2-(√2)^2 = x^2-2. Let us divide p(x) by x^2-2 to find other factors.
2x^2 +7x+23
--------------------------------
x^2-2 | 2x^4-7x^3+19x^2-14x+13
2x^4 -4x^2
------------------------
7x^3 +23x^2-14x
7x^3 -14x
-----------------------
23x^2 +13
23x^2-46
It is seen that 2x^4-7x^3+19x^2-14x+13 is not divisible by x^2-2 so there is something wrong in the question. Please check it