Warm Up (in your spiral) 1. Expand: (x + 3) 2 2. Expand: (x + 1/2) 2 3. Factor: x 2 + 8x Factor: 3x x + 8

Chapter Four Polynomial and Rational Functions

Section 4.2: Quadratic Equations Objectives:  Solve quadratic equations.  Use the discriminant to describe the roots of quadratic equations.

Formula Refreshers…  Quadratic Formula  Discriminant: Zero Real Solutions Two Real Solutions One Real Solution

Example 1: Solve for x: (3x-2)(x+1) = 0 (3x – 2) = 0 or (x + 1) = 0 3x – 2 = 0 or x + 1 = 0 x = 2/3 x = -1

Example 2: Solve for n: …. can’t factor… Quadratic Formula

Discriminant +  Two Real Roots  Two Imaginary Roots  One Real Root (possible Multiplicities) - 0

Example 3: Solve

Example 4: Find the discriminant of and describe the nature of the roots. Then solve the quadratic using the Quadratic Formula.

Warm-Up

Section 4.1: Polynomial Functions Objectives:  Determine roots of polynomial equations.  Apply the Fundamental Theorem of Algebra.

Example 1: Consider the polynomial equation. a) State the degree and leading coefficient of the polynomial. b) Determine whether -2 is a zero of f(x). Degree: 4, Lead Coeff: 3 f(-2) = 0? 3(-2) 4 – (-2) 3 + (-2) 2 -1 f(-2) = 57 No a zero

Discussion time  Review what you discovered/remembered about complex numbers with your partner. Record ANY important facts or bits of information. Complex numbers: a + bi Imaginary Taking the square root of a negative…. So, if you have one, then you ALWAYS have another

Conjugates Sample: If -4i is a root of an equation, then _________ MUST also be a root. a+ bi a – bi +4i

Example 2: Write a polynomial of at least degree of 3 with roots 2, And 3i. Does the polynomial have an odd or even degree? How many times does it cross the x-axis?

Example 3 : State the number of complex roots of. Then find the roots.

Example 4 State the number of complex roots of. Then find the roots.

Example 5: State the number of complex roots of. Then find the roots.

Warm Up ( ) Divide 4323 by 12 BY HAND

Section 4.3: The Remainder and Factor Theorems (Day 1) Objectives:  Use Long Division and Synthetic Division to find Quotients and Remainders.  Find the factors of polynomials using the Remainder and Factor Theorems.

Example 1: Divide x 8 – x + 4 by x using long division.

Example 2: Use synthetic division to divide 5x 4 + 7x 3 – 8x 2 + 2x + 1 by x – 2.

Example 3: Use synthetic division to divide x 3 – x by x +1.

Example 4 : Use synthetic division to divide 2x 3 - 4x – 6 by 2x – 6.

Example 5 (Section 3.7) : Find the Oblique Asymptote for

Warm Up (From 4.3 Day 1) Find the rest of the roots for If x + 5 is a factor.

Section 4.3: The Remainder and Factor Theorems (Day 2)

Theorem – The Remainder Theorem: Theorem – The Factor Theorem: P(x) = Q(x) + c x – r x - r polynomial Possible “Factor” Quotient remainder x – r is a factor of p(x) iff c = 0. (we already knew both of these)…. Asd;flklkajsd; JUST DIVIDE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!qwdlakjds;l

Example 6: Use the Remainder Theorem to find the remainder when x 3 – x 2 – 5x – 3 is divided by x – 3. State whether the binomial is a factor of the polynomial. Explain. Just means divide...

Example 7: Determine the binomial factors of x 3 – 2x 2 – 13x – 10…. Graph the equation to find one of the solutions… x = -2 x = -1 x = 5 Factors: (x + 2) (x + 1) (x – 5) Confirm by using synthetic division with one of the factors: _____________________ Remaining polynomial: x 2 – 4x + 5 = (x – 5) (x + 1)… other 2 factors Remainder of 0 means (x + 2) is a factor!

Example 8: Find the value of j, so that the remainder of x 3 + 6x 2 – jx – 8 is zero, when divided by x j - 8 _____________________ j (-j + 16) To get a remainder of 0, we need 2(-j + 16) to equal 8. 2(-j + 16) = 8 -2j + 32 = 8 -2j = -24 j = 12

Warm Up Find all complex roots of: …and identify them “Complex”, “Real”, “Imaginary” Roots: Complex: Real: Imaginary:

What I expect you to know and be able to do at this point:  How to solve a Quadratic  FACTORING  QUADRATIC FORMULA  Find the discriminant and know what it means  + 2 Real  - 2 Imaginary  0 1 Real  Long Division  Find a root off of a table  Synthetic Division  Use it to Factor Cubics  Rationalize Denominators – we haven’t done this, but you KNOW how to do it…  Simple (Regular) Radicals  Complex Conjugates  Simplify Radicals

Section 4.4: The Rational Root Theorem Objectives:  Identify all possible rational roots of a polynomial equation by using the Rational Root Theorem.  Determine the number of positive and negative real roots a polynomial function has.

Example 1: List the possible rational roots of. Then determine the rational and complex roots. Let p = factors of the “last term”. Let q = factors of the leading coefficient Start with “whole number” terms _____________________ x = -2 is a factor Use 6x 2 – x – 1 to find the remaining factors. (Factor or use Quad Form) x = -2, ½, -1/3

Theorem – Rational Root Thm: P = the factors of c Q = the factors of a Possible roots: Therefore,

Example 2: List the possible rational roots of. Then determine the roots.

Example 3: List the possible rational roots of Then determine the rational roots.

Spartans Will!

In-State College Day  Did you know that MSU is only and hour and 15 minutes away?  That’s far enough time to feel like you’re “free”  But close enough to still have your mommy do your laundry every once and a while!  Did you know there are price breaks for “In State Tuition” ??

Warm-Up: Find the roots of without a calculator

Section 4.6: Rational Equations and Partial Fractions Objectives:  Solve rational equations and inequalities.  Decompose a fraction in to partial fractions.

Example 1: Solve

Example 2: Solve

 Did you know that MSU has programs/affiliates in almost every state of the US?  Did you know that MSU boasts one of the most broad and in- depth “Study Abroad” programs in the nation? Out-of-State College Day

Example 3: Decompose into partial fractions.

3 5

Example 4: Decompose into partial fractions.

4 2

Warm-Up Break into partial fractions:

Example 5: Solve Roots: Domain Restrictions: Test points from each interval -1 is a Domain Restriction, So it acts as a stopper. Cannot write One beautiful interval for that. Where is the function greater Than 0? (Positive)??

Example 6 (Warm-UP): Solve Roots: Domain Restrictions: Test points from each interval 4 is a Domain Restriction, stopper. (Doesn’t really matter anyway….) Where is the function less than 0? (Negative)??

Warm-Up: Decompose  3x+6 x 2 -8x-9

Small College Day  MSU is big.  Campus is set up as a cluster of “Pods” that make you feel like you’re at a smaller, more quaint college.  Like Central/Western/GVSU – Brody Complex: Newest & great food.  Like East Coast Ivy League – North Campus: The most beautiful spots on campus!! Wooded, green, gorgeous.  Like the Sciences/Math: Lyman-Briggs College within a College  There are many Student Life Organizations/activities to get involved in

Warm- Up

Section 4.7: Radical Equations and Inequalities Objectives:  Solve radical equations and inequalities.

We will be doing the notes out of order today, to do easiest to hardest Order: Example 1 Example 4 Example 5 Example 2 – Change Example 3

Example 1: Solve X= 9, 2

Example 4: Solve Now lets talk about the domain restrictions

Example 5: Solve

Example 2 (CHANGE IN NOTES!!!!!) Solve

Example 3: Solve X= 8, 24 Square both sides Simplify Square both sides Simplify

Warm Up (From 4.6) 1. Decompose into partial fractions. 2. Solve

What I expect you to know and be able to do at this point:  How to solve a Quadratic  Find the discriminant and know what it means  Division  Roots  How to factor a Cubic or Quartic  Solving “radical” (square root problems)  Test point method  Finding Common D’s and crossing them out  Partial Fractions and Decomposing them

Problem to add on the back 6. Factor x 3 +3x 2 +3x+1