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

September 17, 2021

By Lyla

1 thought on “Find all other zeros of polynomial x⁴+x³-9x²-3x+8 if it is given two zeros are that<br />[tex] \sqrt{3} and – \sqrt{3} [/tex]<b”

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

   ———————————————

   0+X³ – 6X² – 3X + 18

   X³-3

   —————————————-

   0*-6X² 0*+18

   -6X² +18

   ——————————————

   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.

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