If two zeroes of the polynomial $x^{3} -4x^{2} -3x+12=0$ are $\sqrt{3}$ and $-\sqrt{3}$, then find its third zero.
Given: A polynomial $x^{3} -4x^{2} -3x+12=0$ with two zeroes $\displaystyle \sqrt{3}$ and $ -\sqrt{3}$
To do: To find out its third zero.
Solution:
Here the given polynomial is $x^{3} -4x^{2} -3x+12=0$
For a polynomial $ax^{3} +bx^{2} +cx+d=0$ if the zeroes $\alpha,\beta\ and\ \gamma $.
$\alpha+\beta+\gamma=-b$
Here $a\ =\ 1,\ b=-4,\ c=3\ and\ d=12\ $
And $\alpha=\sqrt{3} ,\ \beta=-\sqrt{3}$ and we have to find the value of third zero $\gamma$.
$\therefore \sqrt{3} -\sqrt{3} +γ=-( -4)$
$\Rightarrow \gamma=4$
Thus the third zero of the given polynomial is 4.
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