Polynomials (C10)

Find all zeroes of the polynomial (2x4 – 9x3 + 5x2 + 3x – 1) if two of its zeroes are (2 + √3) and (2 – √3).

Answer

p(x) = 2x4 – 9x3 + 5x2 + 3x – 1

Since, 2 + √3 and 2 - √3 are the zeroes of p(x).

Therefore (x - 2 - √3) and (x - 2 + √3) are factors of p(x).

p(x) = (x - 2 - √3)(x - 2 + √3) × g(x)

p(x) = (x2 - 4x + 1) × g(x)

Now, divide p(x) by (x2 - 4x + 1) to get g(x)

g(x) = 2x2 - x - 1

g(x) = (x - 1)(2x + 1)

Therefore, other two zeroes are 1, -1/2.