Find the polynomial zeros , Mathematics

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If two zeros of the polynomial f(x) = x⁴ - 6x³ - 26x² + 138x - 35 are 2 ± √3.Find the other zeros. (Ans:7, -5)

Ans: Let the two zeros are 2 +√3 and 2 - √3

Sum of Zeros = 2 + √3 + 2 - √3

= 4

Product of Zeros = ( 2+√3 )(2 - √3 )

= 4 - 3

= 1

Quadratic polynomial is x² - (sum) x + Product

x⁴ - 4 x³ + x²

-----------------

-2x³ - 27x² + 138x

- 2x³ + 8x² - 2x

-----------------------

-35x² + 140x - 35

------------------------

∴ x² - 2x - 35 = 0

(x - 7)(x + 5) = 0

x = 7, -5

other two Zeros are 7 and -5

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