posted by Marissa on .
Can someone please explain how to do these problems.
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
Ths means that p is a divisor of 6 and q a divisor of 3.