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Obtain All Zeros of the Polynomial F(X) = X4 − 3x3 − X2 + 9x − 6, If Two of Its Zeros Are `-sqrt3` and `Sqrt3` - CBSE Class 10 - Mathematics

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Question

Obtain all zeros of the polynomial f(x) = x4 − 3x3 − x2 + 9x − 6, if two of its zeros are `-sqrt3` and `sqrt3`

Solution

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.

  Is there an error in this question or solution?

APPEARS IN

 RD Sharma Solution for 10 Mathematics (2018 to Current)
Chapter 2: Polynomials
Ex. 2.30 | Q: 5 | Page no. 57

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Solution Obtain All Zeros of the Polynomial F(X) = X4 − 3x3 − X2 + 9x − 6, If Two of Its Zeros Are `-sqrt3` and `Sqrt3` Concept: Division Algorithm for Polynomials.
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