1. Zeros of Polynomial Functions
Take out your homework
Warm-up
Solve  3x3 – 2x2 + 3x – 4 ÷ x – 3 by Synthetic Division
Find f(25) using synthetic substitution in the above expression
List all the potential real factors of the expression in #1.
Quiz Review:

2. Warm-up Answers
45,696
±1, ±2, ±4, ±4/3,±2/3, ±⅓

3. Learning Goals Fundamental Theorem of Algebra Conjugate Root Theorem
Complex Zeros
…be able to find a polynomial given zeros, real and complex
…be able to factor and find complex zeros of a polynomial function

4. Fundamental Theorem of Algebra
An nth-degree polynomial (n >0), has at least one zero in the complex number system.
Corollary – An nth-degree polynomial function has exactly n zeros, including repeated zeros, in the complex number system.

5. Conjugate Root Theorem
If a + bi (b ≠ 0), is a root of a polynomial function with real coefficients, then the complex conjugate, a - bi is also a root of the polynomial.

6. Find a Polynomial Function Given Its Zeros
Write a polynomial function of the least degree possible, with real coefficients in standard form with the given zeros.
Pg. 127: #35 (homework problem)
**hint – Conjugate Root Theorem
Solution:
Zeros: -1, 8, (6 – i)

7. Practice (10 min.)
Pg. 127: #32, 33, 38, 85-86
Write the polynomial function in standard form, given the zeros.
3, -4, 6, -1
-2, -4, -3, 5
Use Synthetic Division

8. Wrap-up and Quiz Review
I am able to find a polynomial given zeros, real and complex.
Quiz Friday covering Long Division and Synthetic Division/Substitution.

9. Quiz Review
Take out your Quiz Review Worksheet.

10. Factoring Polynomials
Every polynomial can be written as the product of linear factors and/or irreducible quadratic factors, each with real coefficients.
A quadratic factor is irreducible over the reals when it has real coefficients, but no real zeros associated with it.
Example: (x2+4)

11. Factor and Find the Zeros of a Polynomial Function
Starting point?
Total number of Zeros
Rational Zero Theorem – List all possible Rational Zeros
Graph to find likely real zeros
Use Synthetic Division to:
Check for factors
factor the function

12. Example Factor and find the zeros of the equation: Total zeros – 5
RZT – All possible rational zeros
– {±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30}
Find potential zeros on a graph that are in the list.

13. Choose possible zeros to try with synthetic substitution.
Example continued:
Choose possible zeros to try with synthetic substitution.
(-5, 2, 3 are all good possibilities) -5 1
-18 30
-19 25
-35
-30 7 6 2 1 -5 7 6 -6 -3 3 1 -3
𝑥 2 +1

14. Factor polynomial – all linear factors
Continue from the previous Example:
𝑓 𝑥 = 𝑥+5 𝑥−2 𝑥−3 𝑥 2 +1
Rewrite all irreducible factors as complex linear factors.
Final answer written in all linear factors

15. Wrap-up
I am able to factor and find complex zeros of a polynomial function.

16. Practice Pg. 127: #42-44, Complex Zeros WS
42-44 – write as a) the product of irreducible linear factors, b) the product of linear factors, and c) list all the zeros (Real and Complex)