Find the zeros of the polynomial 2x^4 + 7x^3 - 19x^2 - 14x + 30. If two of its zeros are square root of 2 and. - square root of 2

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Solution

Zeroes of polynomial are √2 and -√2
Factor will be
(x-√2)(x+√2)
=(x²-2)

now, dividing the polynomial with this factor,

x²-2)2x⁴ +7x³ -19x² -14x +30(2x²+7x-15
2x⁴ -4x²
- +
0 7x³-15x²-14x+30
7x³ -14x
- +
0 -15x 0 +30
-15x +30
+ -
0 0

So, the factors are (x²-2)(2x²+7x-15)
=(x²-2)(2x²+10x -3x -15)
=(x²-2)[2x(x+5)-3(x+5)]
=(x²-2)(x+5)(2x-3)

so, other zeroes are -5 and 3/2

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

\sqrt{2}

- \sqrt{2} .

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

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