Question

Obtain all the zeroes of the polynomial f(x) = 4x4 – 16x3 + 13x2 + 12x – 12 if its two zeroes are (underroot3/2 and -underoot3/2)

Answer 1

Answer:

Other two zeroes are 2

Step-by-step explanation:

f(x) = 4x⁴  - 16x³  + 13x² + 12x  - 12

Two zeroes are √3 /2  &  -√3/2

g(x) = (x - √3 /2)(x + √3 /2) = x² - 3/4  = 4x² - 3

f(x) = g(x)p(x)

p(x) = ax² + bx + c

=>f(x) =  (4x² - 3)(ax² + bx + c)

= 4ax⁴ + 4bx³ + x²(4c -3a)   -3bx -3c

comparing with

4x⁴  - 16x³  + 13x² + 12x  - 12

=> 4a = 4 => a = 1

4b = -16 => b = -4

-3c = -12 => c = 4

p(x) = x² - 4x  + 4

= x² - 2x - 2x + 4

= x(x-2) -2(x -2)

= (x-2)(x-2)

= (x - 2)²

Other two zeroes are 2