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Find All Zeros of the Polynomial 2x4 + 7x3 − 19x2 − 14x + 30, If Two of Its Zeros Are `Sqrt2` and `-sqrt2`. - CBSE Class 10 - Mathematics

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Question

Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.

Solution

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

Since, `sqrt2` and `-sqrt2` are zeros of f(x).

Therefore

`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`

= x2 - 2

x2 - 2 is a factor of f(x). Now, we divide 2x4 + 7x3 − 19x2 − 14x + 30 by g(x) = x2 - 2 to find the zero of f(x).

By using division algorithm we have

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

2x4 + 7x3 − 19x2 − 14x + 30 = (x2 - 2)(2x2 + 7x - 15) + 0

2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x^2+10x-3x-15)`

2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)[2x(x+5)-3(x+5)]`

2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x-3)(x+5)`

Hence, the zeros of the given polynomial are `-sqrt2`, `+sqrt2`,  `(+3)/2`, -5.

  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: 10 | Page no. 57

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Solution Find All Zeros of the Polynomial 2x4 + 7x3 − 19x2 − 14x + 30, If Two of Its Zeros Are `Sqrt2` and `-sqrt2`. Concept: Division Algorithm for Polynomials.
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