If two zeroes of the polynomial x⁴ – 6x³ – 26x² + 13.8x – 35 are find other zeroes.

Given:

2+√3 and 2–√3 are zeroes of given equation

P(x) = x⁴ – 6x³ – 26x² + 13.8x – 35
Explanation:
(x – 2 + √3)(x – 2 – √3) should be a factor of the given equation.
Also, (x – 2 + √3)(x – 2 – √3) = x² – 2x – √3x –2x + 4 + 2√3 + √3x – 2√3 – 3
= x² – 4x + 1
To find other zeroes, we divide the given equation by x² – 4x + 1

We get,

x⁴ – 6x³ – 26x² + 13.8x – 35 = (x² – 4x + 1)(x² – 2x – 35)

Now factorizing x² – 2 x – 35 we get,

x² – 2x – 35 is also a factor of given polynomial and x² – 2x – 35 = (x – 7)(x + 5)

The value of polynomial is also zero when,

x – 7 = 0, x = 7

x + 5 = 0, x = –5

Hence, 7 and –5 are also zeroes of this polynomial.