1. 3.4 Zeros of Polynomial Functions: Real, Rational, and Complex
Complex Zeros and the Fundamental Theorem of Algebra
It can be shown that if a + bi is a zero of a polynomial function with real coefficients, then so is its complex conjugate, a – bi.
Conjugate Zeros Theorem
If P(x) is a polynomial having only real coefficients, and if a + bi is a zero of P, then the conjugate a – bi is also a zero of P.

2. 3.4 Topics in the Theory of Polynomial Functions (II)
Example Find a polynomial having zeros 3 and 2 + i that
satisfies the requirement P(–2) = 4.
Solution Since 2 + i is a zero, so is 2 – i. A general solution is
Since P(–2) = 4, we have

3. 3.4 Zeros of a Polynomial Function
Example
Find all complex zeros of
given that 1 – i is a zero.
Solution
Number of Zeros Theorem
A function defined by a polynomial of degree n has at most n distinct complex zeros.

4. 3.4 Zeros of a Polynomial Function
Using the Conjugate Zeros Theorem, 1 + i is also a zero.
The zeros of x2 – 5x + 6 are 2 and 3. Thus,
and has four zeros: 1 – i, 1 + i, 2, and 3.

5. 3.4 The Rational Zeros Theorem
Example
List all possible rational zeros.
Use a graph to eliminate some of the possible zeros listed in part (a).
Find all rational zeros and factor P(x).
The Rational Zeros Theorem

6. 3.4 The Rational Zeros Theorem
Solution
(a)
From the graph, the zeros
are no less than –2 and no
greater than 1. Also, –1 is
clearly not a zero since the
graph does not intersect the
x-axis at the point (-1,0).

7. 3.4 The Rational Zeros Theorem
(c) Show that 1 and –2 are zeros.
Solving the equation 6x2 + x – 1 = 0, we get x = –1/2, 1/3.