Step-by-step explanation:
By the fundamental theorem of algebra, the degree of the polynomial equals the number of roots (including complex and multiplicities, i.e. multiple roots).
Since n=4, we expect to find 4 roots, although we are only given three, namely -1,3,2+4i.
Fortunately, if the polynomial has real coefficients (which we will <em>ASSUME</em>), for any complex root to exist, its conjugate is also a root to the equation.
Since the conjugate of 2+4i is 2-4i, we now have all four roots, -1, 3, 2+4i, 2-4i.
An n-degree polynomial may be construct by the product of (x-ri) where ri are the roots. This means
P(x) = (x-(-1))(x-3)(x-2-4i)(x-2+4i), which is
P(x) = (x+1)(x-3)(x-2-4i)(x-2+4i)
When expanded, P(x) = x^4-6x^3+25x^2-28x-60
Check:
sum of roots = (-1+3+2+4i+2-4i) = 6 (same as negative of coefficient of term x^3) ok
product of roots = (-1)(3)(2+4i)(2-4i)=-3*20 = -60 = constant term, ok.
Note: kP(x) is also a polynomial with the same four roots, where k=any real number. P(x) is a particular case where k=1.
