b Find a polynomial with integer coefficients and a leading

b) Find a polynomial with integer coefficients and a leading coefficent of one which satisfies the given conditions

Find a polynomial with integer coefficients and a leading coefficent of one which satisfies the given conditions P has degree 4 and zero 2 + i and squareroot 3.

Solution

zeros are 2+i and sqrt 3

complex zeros occur in pairs so if 2+i is one zero other would be 2-i

similarly if sqrt 3 is one zero other would be - sqrt 3

we can express polynomial as

p(x) = (x-a) (x-b) (x-c)(x-d)

where , a,b,c,d are the zeros of the polynomial

p(x) = (x- (2+i)) (x-(2-i)) (x-sqrt 3) (x+sqrt 3)

p(x) = (x-2-i)(x-2+i)(x-sqrt 3)(x+sqrt 3)

p(x) = (x^2-4x+5)(x^2 - 3)

so the polynomial of degree 4 is

p(x) = x^4 - 4x^3 + 2x^2 + 12x - 15