Question

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

2x⁴ – 9x³ + 5x² + 3x – 1; 2 ± √3

Answer 1

2x⁴ – 9x³ + 5x² + 3x – 1;2±√3

Given zeroes are 2 + √3 and 2 – √3

So, (x – 2 – √3)and (x – 2 + √3) are the factors of 2x⁴ – 9x³ + 5x² + 3x – 1

⟹ (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 2x⁴ – 9x³ + 5x² + 3x – 1 by x² – 4x + 1

The division process is

Here, quotient = 2x² – x – 1

= 2x² – 2x + x – 1

= 2x(x – 1) + 1(x – 1)

= (2x + 1)(x – 1)

So, the zeroes are -1/2 and 1

Hence, all the zeroes of the given polynomial are -1/2, 1, 2 + √3 and 2 – √3