Question

find all the zeros of the polynomial 2 x^ 4 + 7 x ^3 - 19 x ^2 - 14 x + 30 if two of its zeros are √2 and -√2 ​

Answer 1

Two zeroes are :- √2 and -√2

(x - √2)(x + √2)

x² + √2x - √2x - 2

x² - 2

•°• (x² - 2) is a factor of p(x)

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Now, divide p(x) by (x² - 2)

(Refer attachment for division)

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To find other zeroes of p(x), we factorise the quotient, or use the quadratic formula.

Solving by quadratic formula,

$\sf \: x \: = \frac{ - b \: \pm \: \sqrt{ {b}^{2} - \: 4ac } }{2a} \\ \sf \: x \: = \frac{ - 7 \: \pm \: \sqrt{ {7}^{2} - \: 4(2)( - 15) } }{2(2)} \\ \sf \: x \: = \frac{ -7 \: \pm \: \sqrt{ 49 + \: 120 } }{4} \\ \sf \: x \: = \frac{ - 7 \: \pm \: \sqrt{169} }{4} \\ \sf \: x \: = \frac{ - 7 \: \pm \: 13}{4} \\ \sf \: x \: = \: \frac{ - 7 + 13}{4} ,\: x \: = \frac{ - 7 - 13}{ 4 } \\ \sf \: x \: = \frac{6}{4} , \: x = \frac{{ - 20}{} }{4} \\ \sf \: x \: = \frac{3}{2} , - 5$

•°• Other zeroes are 3/2 and -5