I have to find a polynomial with integer coefficients that satisfies the given conditions. thank you!!

Question

U has degree 5, zeros 1/2 , −2, and −i, and leading coefficient 4; the zero −2 has multiplicity 2.

S has degree 4 and zeros 2i and 3i.

Q has degree 3 and zeros 5, 4i, and −4i.

I'm so stuck.

Answer 1

I will work the first one for you, and you work the others the same way.

Degree 5, zeros 1/2, -2 (multiplicity 2), -i, leading coefficient 4

The complex conjugate of a complex zero must also be a zero, so your 5 zeros are

1/2

-2 (twice)

-i

i

U(x) = A(x - 1/2)(x + 2)²(x + i)(x - i)

A = 4

U(x) = 4(x - 1/2)(x + 2)²(x + i)(x - i) = 4(x - 1/2)(x + 2)²(x² + 1)

You can expand this further if you wish. Now try to work the other problems the same way.