Question 3.3.5: Finding a Polynomial Function That Satisfies Given Condition...
Finding a Polynomial Function That Satisfies Given Conditions (Complex Zeros)
Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros 3 and 2 + i.
The complex number 2 – i must also be a zero, so the polynomial has at least three zeros: 3, 2 + i, and 2 – i. For the polynomial to be of least degree, these must be the only zeros. By the factor theorem there must be three factors: x – 3, x – (2 + i), and x – (2 – i).
ƒ(x) =(x – 3)[x – 12 + i2][x – (2 – i)] Factor theorem
ƒ(x) =(x – 3)(x – 2 – i)(x – 2 + i) Distribute negative signs.
ƒ(x) = (x – 3)(x² – 4x + 5) Multiply and combine
like terms; i² = -1.
ƒ(x) = x³ – 7x² + 17x – 15 Multiply again.
Any nonzero multiple of x³ – 7x² + 17x – 15 also satisfies the given conditions on zeros. The information on zeros given in the problem is not sufficient to give a specific value for the leading coefficient.