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Answer 1

Solution :

The two zeroes are (2 ± √3)

Then, the two factors of the given polynomial be

{x - (2 + √3)} and {x - (2 - √3)}

∴ a factor can be

{x - (2 + √3)} {x - (2 - √3)}

= {(x - 2) - √3} {(x - 2) + √3}

= (x - 2)² - (√3)²

= x² - 4x + 4 - 3

= x² - 4x + 1

Now, x⁴ - 6x³ - 26x² + 138x - 35

= x⁴ - 4x³ + x² - 2x³ + 8x² - 2x - 35x²

+ 140x - 35

= x² (x² - 4x + 1) - 2x (x² - 4x + 1)

- 35 (x² - 4x + 1)

= (x² - 4x + 1) (x² - 2x - 35)

∴ the other factor is

= x² - 2x - 35

= (x - 7) (x + 5)

Hence, the other two zeroes are

7 and (- 5)