Question

18. If two zeroes of the polynomial
$(2x {}^{3 } - 4x - x {}^{2} + 2) \: are \\ \sqrt{2} \: and \: - \sqrt{2} \: then \: obtain \: \\ the \: third \: zero.$

Standard:- 10

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Answer 1

Given Polynomial is f(x) = 2x^3 - 4x - x^2 + 2.

Given that root 2 and - root 2 are the zeroes of the polynomial.

$= \ \textgreater \ (x + \sqrt{2} )(x - \sqrt{2} )$

$= \ \textgreater \ x^2 - 2$

Now,

We have to divide the given polynomial by x^2 - 2 to find the factors:



               2x - 1.
             --------------------------
x^2 - 2) 2x^3 - x^2 - 4x + 2

             2x^3          - 4x

            ----------------------------

                          - x^2  +     2

                           - x^2   +   2

              ------------------------------

                                  0.


Now,

= > 2x - 1 = 0

= > x = 1/2


Therefore the third zero of the polynomial is 1/2.


Hope this helps!

Answer 2

Hey There!


The logic is this:

x = ± √2 are zeros.

That means both (x+√2) and (x-√2) are factors.

So, (x+√2)(x-√2) = x² - 2 is also a factor.


The solution is shown in the image.

Hope it helps,
Purva
Brainly Community