If three zeroes of a polynomial x4-x3-3 x2-3 x are 0- -3 and -3 then find the fourth zero-

Let P(x) = x^4 x^3 3x^2 + 3x Given, 0, √(3), √(3) are three zeroes, so x = 0, x = √(3) and x = √(3) ⇒ (x √(3)) = 0 and x + √(3) = 0 Therefore x,(x + √ ...

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Standard X

Mathematics

Zeroes of a Quadratic Polynomial

If three zero...

Question

If three zeroes of a polynomial $x^{4} - x^{3} - 3 x^{2} + 3 x$ are 0, $\sqrt{3}$ and $- \sqrt{3}$ then find the fourth zero.

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Solution

Let $P (x) = x^{4} - x^{3} - 3 x^{2} + 3 x$
Given, $0, \sqrt{3}, - \sqrt{3}$ are three zeroes, so
$x = 0,$
$x = \sqrt{3}$ and $x = - \sqrt{3}$
$\Rightarrow (x - \sqrt{3}) = 0$ and $x + \sqrt{3} = 0$
Therefore $x, (x + \sqrt{3}) a n d (x - \sqrt{3})$ are factors of P(x)
Here, $x (x + \sqrt{3}) (x - \sqrt{3})$ will also be the factor of P(x).
Or, $x (x^{2} - 3)$ will also be the factor of P(x).
then $x^{3} - 3 x) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{4} - x^{3} - 3 x^{2} + 3 x (x - 1$
$x^{4} - 3 x^{2} - + ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - x^{3} + 3 x - x^{3} + 3 x + - - ----------- - 0$
quotient = (x - 1)
So fourth zero $\Rightarrow x - 1 = 0$
x = 1
Hence four zeroes will be $1, 0, \sqrt{3}, - \sqrt{3} .$

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If three zeroes of a polynomial x4−x3−3x2+3x are 0, √3 and −√3 then find the fourth zero.

If three zeroes of a polynomial $x^{4} - x^{3} - 3 x^{2} + 3 x$ are 0, $\sqrt{3}$ and $- \sqrt{3}$ then find the fourth zero.