PX x42x32x26x15 Given x23 is a factor of px find all zeros

P(X) = x⁴+2x³+2x²-6x-15

Given x²-3 is a factor of p(x), find all zeros of the polynomial.

Solution

P(x) = x^4 +2x^3 +2x^2 -6x -15

given x^2 -3 is a factor of P(x)
so P(x) = (x^2 -3 ) (x^2+2x+5)
so now we have to find the roots of x^2 +2x +5 also
-2 + sqrt(4 - 2*5)/2 and -2 - sqrt(4 - 2*5)/2
[-2 +sqrt(-16)] /2 and [-2 - sqrt(-16)] /2
[ -2 +4i ]/2 and [ -2 -4i ]/2
[-1 +2i] and [-1 -2i]

p(x)= (x^2 -3)  (x -(-1+2i)) (x-(-1 -2i))

P(x) = (x +sqrt(3) ) (x- sqrt(3) ) (x -(-1+2i)) (x-(-1 -2i))