Question

find the third zero of the polynomial x cube minus 4 x square - 3 x + 12 if two of its zeros are under root 5 and minus under root- 3
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Answer 1

GIVEN :

Given polynomial is $x^3-4x^2-3x+12$

And given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$

TO FIND :

The third zero for the given polynomial .

SOLUTION :

Given that $x^3-4x^2-3x+12$

To find the zeros we have to equate the given polynomial to zero

$x^3-4x^2-3x+12=0$

By using the Synthetic Division Method we can solve this cubic equation.

4_|   1     -4     -3       12

       0     4      0      -12

_________________

       1      0       -3      0

∴ x-4 is a factor of the given polynomial.

x-4=0

Hence x=4

∴ x=4 is a zero of $x^3-4x^2-3x+12=0$

We have the quadratic equation $x^2-3=0$

$x^2=3$

$x=\pm \sqrt{3}$

∴ $x=\sqrt{3}$ and $x=-\sqrt{3}$ are also the zeros of the given polynomial.

Since the zeros $x=\sqrt{3}$ and $x=-\sqrt{3}$ are given so that the third zero is 4.

∴ The third zero for the given polynomial $x^3-4x^2-3x+12=0$ is 4