Find a polynomial equation with integral coefficients that h

Find a polynomial equation with integral coefficients that has the given numbers as solutions and the indicated degree. (Objective 4. Give your answer in the form anxn + an 1xn 1 + ... + a1x + a0 = 0. )

1, 2 + 3i; degree 3 and please explain how you got the answer. Thanks

Solution

Let the polynomial be P(x)

zeros are 1 , 2 +3i

for 2+3i there is a corresponding complex conjugate 2-3i also a root

So, P(x) = k*(x-1)(x - 2-3i)(x -2 +3i)

where k is constant

Now P(x) =0

So,k*(x-1)(x - 2-3i)(x -2 +3i)

k { (x-1)( x^2 -2x + 3ix -2x +4 - 6i -3ix+ 6i +9 )

= k (x-1)(x^2 -4x +13)

=k (x^3 -4x^2+13x - x^2+4x -13)

= k(x^3 - 5x^2 +17x -13)

Now , k(x^3 - 5x^2 +17x -13) =0

x^3 - 5x^2 +17x -13 =0 This is the final form of polynomial