Obtain all zeros of the polynomials p(x x4 3x3 x 2 9x 6 if two of its zeros are − 3 and 3)

Try the new Google Books

Check out the new look and enjoy easier access to your favorite features

123456789101112131415161718192021222324252627282930

Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts

Dedicated counsellor for each student

Detailed Performance Evaluation

>

Obtain all zeros of the polynomial fx=x4 3 x3 x2+9 x 6, if two of its zeros are 3 and 3

we know that, if x = a is a zero of a polynomial, then x - a is a factor of f(x).

since `-sqrt3` and `sqrt3` are zeros of f(x).

Therefore

`(x+sqrt3)(x-sqrt3)=x^2+sqrt3x-sqrt3x-3`

= x2 - 3

x2 - 3 is a factor of f(x). Now , we divide f(x) = x4 − 3x3 − x2 + 9x − 6 by g(x) = x2 - 3 to find the other zeros of f(x).

By using that division algorithm we have,

f(x) = g(x) x q(x) + r(x)

x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)(x2 - 3x + 2) + 0

x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)(x2 - 2x + 1x + 2)

x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)[x(x - 2) - 1(x - 2)]

x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)[(x - 1)(x - 2)]

x4 − 3x3 − x2 + 9x − 6 `= (x - sqrt3)(x+sqrt3)(x-1)(x-2)`

Hence, the zeros of the given polynomials are `-sqrt3`, `sqrt3`, +1 and +2.