Find all the zeroes of the polynomial $2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7$ .It being given two of its zeroes are $(3 + \sqrt{2})$ and $(3 - \sqrt{2}) .$

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Solution

Let $α$ and $β$ be zeroes of polynomial
$∴ α = 3 + \sqrt{2}$ and $β = 3 - \sqrt{2}$
$∴ α + β = 6$ and $α \cdot β = 7$
$∴ x^{2} - 6 x + 7$
Given polynomial is $2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7$
$∴ x^{2} - 6 x + 7) 2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7 (2 x^{2} + x - 1$
$- 2 x^{4} - 12 x^{3} + 14 x^{2}$
___________
$x^{3} - 7 x^{2} + 13 x - 7$
$x^{3} - 6 x^{2} + 7 x$

$- x^{2} + 6 x - 7$
$- x^{2} + 6 x + 7$
_
$0$
___________
$∴$ The Quotient is $2 x^{2} + x - 1$
$= 2 x^{2} + 2 x - x - 1$
$= 2 x (x + 1) - 1 (x + 1)$
$= (2 x - 1) (x + 1)$
$∴$ $x = \frac{1}{2}, - 1$ are the other zeros of the polynomial.

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Find all the zeroes of the polynomial 2x4−11x3+7x2+13x−7 .It being given two of its zeroes are (3+√2) and (3−√2).

Find all the zeroes of the polynomial $2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7$ .It being given two of its zeroes are $(3 + \sqrt{2})$ and $(3 - \sqrt{2}) .$