Question

Find all the zeroes of the polynomial given below having given numbers as its zeroes.

x⁴ – 6x³ – 26x² + 138x – 35; 2 ± √3

Answer 1

x⁴ – 6x³ – 26x² + 138x – 35;2±√3

Given zeroes are 2 + √3 and 2 – √3

So, (x – 2 – √3)and (x – 2 + √3) are the factors of x⁴ – 6x³ – 26x² + 138x – 35

⟹ (x – 2 – √3)(x – 2 + √3)

= x² – 2x + √3 x – 2x + 4 – 2√3 – √3 x + 2√3 – 3

= x² – 4x + 1 is a factor of given polynomial.

Consequently, x² – 4x + 1 is also a factor of the given polynomial.

Now, let us divide x⁴ – 6x³ – 26x² + 138x – 35 by x² – 4x + 1

The division process is

Here, quotient = x² – 2x – 35

= x² – 7x + 5x – 35

= x(x – 7) + 5(x – 7)

= (x + 5)(x – 7)

So, the zeroes are – 5 and 7

Hence, all the zeroes of the given polynomial are – 5, 7, 2 + √3 and 2 - √3.