Question

Find all other zeros of polynomial x⁴+x³-9x²-3x+8 if it is given two zeros are that
$\sqrt{3} and - \sqrt{3}$


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Answer 1

Heya !!!

• ( root 3 ) and (- root 3) are the two zeroes of the given polynomial.

• ( X - root 3 ) ( X + root 3 ) are also factor of polynomial P(X).
• Therefore,

( X - root 3 ) ( X + root 3) = (X² - 3)

G(X) = X²-3

P(X) = X⁴ + X³ - 9X² - 3X+ 18

• On dividing P(X) by G(X) we get,

• X² - 3 ) X⁴ + X³ - 9X² - 3X + 18 ( X² + X -6*X⁴ -3X²

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0+X³ - 6X² - 3X + 18

X³-3

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0*-6X² 0*+18

-6X² +18

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We get,

Remainder = 0

And,

Quotient = X² + X - 6

• After factorise the quotient we will get two other zeroes of the given polynomial.

=> X²+X -6

=> X² + 3X - 2X -6

=> X ( X + 3) - 2 ( X +3)

=> (X + 3) ( X -2) = 0

=> (X + 3) = 0 OR (X -2) = 0

=> X = -3 OR X = 2

• Hence,-3 , root 3 , 2 and - root 3 are four zeroes of the polynomial X⁴+X³-9X² -3X + 18.