If two zeros of the polynomial f(x) = x^{4} - 6x^{3} - 26x^{2} + 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 x^{2} - (sum) x + Product
x^{4} - 4 x^{3} + x^{2}
-----------------
-2x^{3} - 27x^{2} + 138x
- 2x^{3} + 8x^{2} - 2x
-----------------------
-35x^{2} + 140x - 35
------------------------
0
∴ x^{2} - 2x - 35 = 0
(x - 7)(x + 5) = 0
x = 7, -5
other two Zeros are 7 and -5