Question

Find the nth degree polynomial with real coefficients satisfying the given conditions.

n=3, 4 and 2i are zeros, f(-1)=50

The expanded and simplified polynomial is f(x)=_________________

Answer 1

f(x) = x³ -4x²- 12x +80

Answer 2

that is wrong

Answer 3

The degree is 3, so you have a polynomial of the form
\[ax^3+bx^2+cx+d\]
You know the roots are \(x=4\) and \(x=\pm2i\) (complex roots come in conjugate pairs), so you can write it as
\[a(x-4)(x-2i)(x+2i)\]
You know that \(f(-1)=50\), so
\[a(-1-4)(-1-2i)(-1+2i)=50~~\Rightarrow~~a=-2\]
So, the polynomial (unexpanded) is
\[f(x)=-2(x-4)(x-2i)(x+2i)\]