Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1and the zeros - 3 and 5 + i

Answer 1

9514 1404 393

Answer:

f(x) = x³ -7x² -4x +78

Explanation:

The complex zeros come in conjugate pairs. Each zero (p) gives rise to a factor of the form (x -p). The polynomial is ...

f(x) = (x -(-3))(x -(5+i))(x -(5-i)) = (x +3)((x -5)² -i²)

= (x +3)(x² -10x +25 +1) = (x³ -10x² +26x) +(3x² -30x +78)

f(x) = x³ -7x² -4x +78