Can someone please explain how to do these problems.

Question

1)write a polynomial function of least degree with intregal coefficients whose zeros include 4 and 2i.

2)list all of the possible rational zeros of f(x)= 3x^3-2x^2+7x+6.

3)Find all of the rational zeros of f(x)= 4x^3-3x^2-22x-15.

Because all the coefficients of the polynomial are real, if 2i is a zero, then the complex conjugate -2i must also be a zero. This means that the least order polynomial is proportional to:

(x-4)(x-4i)(x+4i)= (x-4)(x^2 + 16)=
x^3 - 4x^2 + 16 x - 64

To find the rational roots, use the Rational Roots Theorem: if x = p/q is a rational root (and p and q don't have any divisors in common), then p must divide the constant term and q the coefficient of the highest power of x.

In case of the polynomial

3x^3-2x^2+7x+6.

Ths means that p is a divisor of 6 and q a divisor of 3.

Answer 1

The possible rational roots are therefore:

p/q = ±1, ±2, ±3, ±6

For the polynomial

4x^3-3x^2-22x-15

The possible rational roots are:

p/q = ±1, ±3, ±5, ±15