Unit 3, Polynomial Functions: Day 13 (BI, Chapter , Section 6, Day 1 )

Focus: Fundamental Theorem of Algebra

Action Items:

1. Warm Ups

2. Show me HW 3.13

3. Notes and practice with the Rational Root Theorem (including working backwards from solutions to factors and complex conjugates)

4. HW 3.14: BI p. 202 #17-20 all; and #21-29 odd. As always Write The Problems and Show Your Steps! Note: if the homework does not make sense when you are working on your own, do not simply "quit." Instead of completing the assigned problems watch one of the videos in the Polynomials Unit Resources and annotate it (take notes and copy examples) as an alternate assignment.

Goals for today:

Know what the Fundamental Theorem of Algebra states
Be able to use the FTA to aid in solving and generating polynomial equations
Understand and be able to identify and generate Complex Conjugates and Irrational Conjugates
Be able to use binomial multiplication to simplify conjugates
Continue practice with Factoring (all sorts) & Solving

Review of recent topics (see prior days or resources):

1. Rational Root Theorem

2. Process for solving polynomials:

make sure the polynomial is in standard form
set the polynomial equal to zero
factor --> set each factor equal to zero
use ZPP (zero product property) to solve for x-values (the solutions/roots)
remember that x-intercepts occur at the zeros of the polynomial

3. Process for Factoring Polynomials:

GCF! Always check for a greatest common factor first -- and not just the numbers! Always check to see if there is an "x" in every term!
Special Products:

- Difference of Squares (DoS) and Sums & Differences of Cubes (SoC and DoC 😉)
- Perfect Square Trinomials (higher degrees)

3. Apply the "Quadratic Form (Structure)" to polynomials of higher (even) degree

Example: you can factor $LaTeX: x^2+7x+12=\left(x+3\right)\left(x+4\right)$ --> --> so you can also factor $LaTeX: x^4+7x^2+12=\left(x^2+3\right)\left(x^2+4\right)$ . Look carefully to see what is the same and be sure you understand why!

4. Factoring by Grouping

5. Don't stop until the polynomial is FULLY FACTORED! (Try for all linear factors -- be on the lookout, especially, for DoS or DoC if you have non-linear binomials!)

HW 3.14: BI p. 202 #17-20 all; and #21-29 odd. As always Write The Problems and Show Your Steps!

Polynomials Unit Resources

Polynomials Unit Homework Solutions

Don't forget you have access to Tutor.com: 24/7 -- the link is on the left!