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If two zeroes of the polynomial x^4 – 6x^3 – 26x^2 + 138x – 35 are 2 ± root 3, find other zeroes.

Q4   If two zeroes of the polynomial x^4 - 6x ^3 - 26 x^2 + 138 x - 35x^4 - 6x ^3 - 26 x^2 + 138 x - 35                are  2 \pm \sqrt 3 , find other zeroes.

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S safeer

Given the two zeroes are

 2+\sqrt{3}\ and\ 2-\sqrt{3}

therefore the factors are

[x-(2+\sqrt{3})]\ and[x-(\ 2-\sqrt{3})]

We have to find the remaining two factors. To find the remaining two factors we have to divide the polynomial with the product of the above factors

[x-(2+\sqrt{3})]\[x-(\ 2-\sqrt{3})]=x^2-(2+\sqrt{3})x-(\ 2-\sqrt{3})x+(\ 2+\sqrt{3})(\ 2-\sqrt{3})\\=x^2-2x-\sqrt{3}x-2x+\sqrt{3}x+1\\=x^2-4x+1

Now carrying out the polynomial division

Now we getx^2-2x -35 \ is \ also \ a\ factor

\\x^2-2x -35 =x^2-7x+5x-35\\=x(x-7)+5(x-7)\\=(x-7)(x+5)

So the zeroes are 2+\sqrt{3}\ ,\ 2-\sqrt{3},7\ and\ -5

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