Question

Obtain all zeros of the polynomial $f(x)= x^{4} -3x^{3}-x^{2}+9x-6$ if two of its zeros are $-\sqrt{3} and \sqrt{3}$

Answer 1

Method of finding the remaining zeros of a polynomial when some of its zeros are given:

We firstly write the factor of polynomial using given zeros and multiply them to get g(x). Then divide a given polynomial by g(x).

The quotient so obtained give other zeros of given polynomial and we factorise it to get other zeros.

SOLUTION:

Let f(x) = x⁴ -3x³  - x²  + 9x - 6

Given : Two Zeroes of the polynomial f(x) are - √3 & √3. Therefore , (x + √3) & (x - √3) are the two factors of given Polynomial f(x).

(x + √3) (x - √3)  = x² - (√3)²

= x² - 3

[(a+b)(a - b) = a² - b² ]

x² - 3 is a factor of given Polynomial f(x)

Now, Divide f(x) = x⁴ -3x³  - x²  + 9x - 6by g(x) = x² - 3  

[DIVISION IS IN THE ATTACHMENT.]

Hence , all the zeroes of the given Polynomial are: (- √3), (√3), 3 ,1, 2

HOPE THIS ANSWER WILL HELP YOU …..

Answer 2

Given Two Zeroes of the polynomial f(x) are V3 & V3.



Therefore, (x + V3) & (x - V3) are the two factors of given Polynomial f(x). (x 3) (x - V3) =x2 -(3)2 x2 - 3



[(a+b)(a - b) = a2-b]

x2 - 3 is a factor of given Polynomial f(x) Now, Divide f(x) -2x3 x2 - 6x - 3 by g(x) x2-3


[DIVISION IS IN THE ATTACHMENT.] Hence, all the zeroes of the given Polynomial are: (3), V3), -1/2