1. 3.4 Zeros of Polynomial Functions
Review of zeros
A ZERO is an X-INTERCEPT
Multiple Zeros
the zeros are x = 5 (mult of 2), -2 (mult of 3), and -4 (mult of 1)
A polynomial function of degree n has at most n zeros, where each zeros of multiplicity k is counted k times.

2. The Factor Theorem
A polynomial function P(x) has a factor (x – c) if and only if P(c) = 0. Determine whether the given binomial is a factor of P(x). Use synthetic division. If the remainder is zero, the binomial is a factor. Otherwise, the binomial is not a factor.

3. Rational Zero Theorem If
then the possible rational zeros are the factors of
Example: Use the Rational Zero Theorem to list possible rational zeros for the polynomial function
Possible Zeros =

4. Practice
Use the Rational Zero Theorem to list possible rational zeros for the polynomial function
Find the zeros of the function.
Use the list generated above to find the actual zeros.
Use the graphing calculator and synthetic
division to find the zeros.

5. Fundamental Theorem of Algebra
If P(x) is a polynomial function of degree n > 1, then p(x) has at least one complex zero.
Remember, complex numbers are a + bi!
If P(x) is a polynomial function of degree n > 1, then P(x) has exactly n complex zeros.
Find the zeros. (next slide)

6. Example Find the zeros of P(x) = x4 – 4x3 +8x2 – 16x + 16
Because the degree is 4, there are exactly 4 zeros.
To find them, first list the possible rational zeros.
+1, 2, 4, 8, 16 so there are 10 possible rational zeros!
Use the graphing calculator to find the first zero.
It looks like it touches at 2, so we use synthetic division to test.

7. Practice Find all the zeros of the polynomial function.
List the possible rational zeros.
Use the graphing calculator to
narrow the list.
Use synthetic division to reduce
the polynomial and solve.
The three zeros are 5, 4 + 3i, and 4 – 3i.

8. Conjugate Pair Theorem
If a + bi is a complex zero of a polynomial function, then the conjugate a – bi is also a complex zero.
Use the given zero to find the remaining zeros of the function.
Use synthetic division and the given conjugate pair to find the other zeros.

9. Find the Zeros
Because 5 + 3i is a zero, 5 – 3i is also a zero – Conjugate Pair Theorem – so use synthetic division with both. So the zeros are -1/3, 5 + 3i, and 5 – 3i.

10. Find a Polynomial Given Zeros
Find a polynomial function of lowest degree
with integer coefficients that has the given zeros: 4, -3, 2.
(x – 4)(x + 3)(x – 2) 1. Write the zeros as linear factors.
f(x) = x3 – 3x2 + x Multiply.
Zeros: 3, 2i, -2i Find the polynomial of lowest degree
with integer coefficients that has the
given zeros.
f(x) = x3 – 3x2 + 4x - 12

11. Assignment
3.4 page – 50 by 5, skip 35