Unit 3, Polynomial Functions: Day 11 (BI, Chapter 4, Section 5)

Focus: Factoring & Solving Polynomials Using the Rational Root Theorem

Action Items:

1. Warm Ups

2. Show me HW 3.9

3. Notes and practice with the Rational Root Theorem (see notes below).

4. HW 3.11: BI p. 194 #1-3; 5-23 odd; use Desmos or GC to graph. (Chapter 4, Section 5, Day 1). As always WTP and SYS.

Goals for today:

Understand and be able to use the Rational Root Theorem to expedite factoring polynomials
Continue practice with Factoring (all sorts) & Solving

The Rational Root Theorem:

If a polynomial has integer coefficients, then every rational solution of $LaTeX: f\left(x\right)\:=\:0$ has the following form:

$LaTeX: \frac{p}{q}\:=\:\frac{factor\:of\:the\:constant}{factor\:of\:the\:leading\:coefficient}$

What does this mean?

If you have a polynomial $LaTeX: y\:=\:3x^5+\:7x^4-2x\:+8$

The possible integer factors could ONLY be 1, or 2, or 4, or 8. . . or they could be $LaTeX: \frac{1}{3},\:or\:\frac{2}{3},\:or\:\frac{4}{3},\:or\:\frac{8}{3}$

Why does this matter?

If you are trying to factor a polynomial, you don't have to test every number using synthetic division -- you can focus on only the numbers that might work! If none of the integer factors work, there are no rational roots for the polynomial.

Review of recent topis:

1. 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

2. 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!)

Homework 3.11: BI p. 194 #1-3; 5-23 odd; use Desmos or GC to graph. (Chapter 4, Section 5, Day 1).

Polynomials Unit Resources

Polynomials Unit Homework Solutions

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