Find all zeros of the polynomial $2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30,$ if two of its zeros are $\sqrt{2}$ and $- \sqrt{2} .$

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Solution

Polynomial : $2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30$
$\sqrt{2}$ and $- \sqrt{2}$ are the zeroes of the polynomial
$\Rightarrow x^{2} - 2$ divides the given polynomial

$x^{2} - 2) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30 (2 x^{2} + 7 x - 15 2 x^{4} - 4 x^{2} - +____________________7 x^{3} - 15 x^{2} - 14 x + 30 7 x^{3} - 14 x - +_- 15 x^{2} + 30 - 15 x^{2} + 30 + -_____________________x$

$2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30 = (x^{2} - 2) (2 x^{2} + 7 x - 15)$
Solving, $2 x^{2} + 7 x - 15 = 0$ we get other two roots given polynomial.
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$x = \frac{- 7 \pm \sqrt{49 + 120}}{4} = \frac{- 7 \pm \sqrt{169}}{4}$
$x = \frac{- 7 + 13}{4}, \frac{- 7 - 13}{4}$
$x = \frac{3}{2}, - 5$
All zeroes of polynomial are : $\sqrt{2}, - \sqrt{2}, 3 / 2, - 5$

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Similar questions

\sqrt{2} and - \sqrt{2}

p (x) = 2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30 a r e \sqrt{2} a n d - \sqrt{2},

\sqrt{2}

\sqrt{- 2}

2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30 are \sqrt{2} and - \sqrt{2},

2 x^{4} + 7 x^{3} - 19 x^{2} - 14 x + 30

\sqrt{2}

- \sqrt{2}

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