# PreCalc Chapter 3

Ch 3 Polynomial and Rational Functions

Ch 3.1 Polynomials Function of Degree Greater Than 2

Definition of Intermediate Value Theorem ;
Using the Intermediate Value Theorem to Show that f(x)=2x4+3x²-2 Has a Zero Between ½ and ¾;
Finding the Values of x such that f(x)>0 and f(x)<0 and Sketching the Graph of f(x)x4+3x³-4x²

Ch 3.2 Properties of Division

Using Long Division to Divide x4-16 by x²+3x+1 ;
Is x²+3x+1 a Factor of x4-16?
Finding  if f(2) if f(x)=x³-3x²+x+5

Definition of Remainder Theorem ;
Definition of Factor Theorem ;
Finding the Zeros of f(x)=x²-6x+8 by Factoring ;
Finding the Function that has the Zeros of 2, -1, and 3 ;
Using Synthetic Division to Find f(2) given f(x)=x³-3x²+x+5 ;
Determining if 3 is a Zero of f(x)=4x³-9x²-8x-3

Using Synthetic Division to Find f(3), f(-4), f(2-√2) given f(x)=x²-4

Using Synthetic Division to Divide x³+7x²+7x-15 by x+5

Ch 3.3 Zeros of Polynomials

Theorem of Zeros of Polynomials ;
Finding f(x) whose Only Zeros are 2, -1, and 3 and f(1)=5 ;
Zero of Multiplicity ;
Finding the Roots of -2, 0, 1, and finding Multiplicities at each Root of a Function from the Graph ;
Graphing  y=1/64(x-3)(x+4)³(x+1)², finding Multiplicities and y–intercept

Finding the Zeros and their Multiplicities for f(x)=x4-10x³+37x²-60x+36 given that One Zero is x=2 and Another Zero is x=3

Using Descartes Rule of Signs for the following Functions f(x)=3x³-4x²+5x-1

f(x)=3x5-4x²+x-3

Using Descartes Rule of Signs to find Possible Roots of 2x5-x³+x²+3x-4=0 ;
Using First and Second Theorem on Bounds to Find Upper and Lower Bound of 2x³-5x²+4x-8=0
(Part 1)

Using First and Second Theorem on Bounds to Find Upper and Lower Bound of 2x³-5x²+4x-8=0
(Part 2) ;
Making a Chart if 3 Positive Roots Possible and 4 Negative Roots Possible , Degree of 7 ;
Writing f(x)=x4-9x³+22x²-32 as a Product of Linear Factors given 4 is a Zero of Multiplicity of 2

Finding the Zeros of the following Functions using the TI-83/84 Calculator f(x)=x³+x²-6x

f(x)=x5+12x4-83x³-200x²+95x+205

Finding the Zeros of the following Functions using the TI-85/86 Calculator f(x)=x³+x²-6x

f(x)=x5+12x4-83x³-200x²+95x+205

Finding the Zeros of the following Functions using the TI-89 Graphing Calculator
f(x)=2x²-5x-6
f(x)=x6-3.2

Determining the End Behavior of the following Polynomials
f(x)=4x4+2x³-6x+1;
g(x)=-x7+2x5-6

Determining the Behavior of the Polynomial h(x)=(x-3)4(x+2)5(x+4) at a Zero

Ch 3.4 Complex and Rational Zeros of Polynomials

Finding a Quadratic Function with Complex Zeros of 2i

Finding a Quadratic Function with Complex Zeros of 2-3i

Finding a Polynomial Function of Degree of 3 with Zeros of -6 and i

Finding a Polynomial Function of Degree of 3 with Zeros of 4 and 2+3i

Finding a Polynomial Function of Degree of 4 with Zeros of 3, -1  and 4+2i

Finding a Function of Degree of 4 with Zeros of 3+5i and -1-i ;
Finding the Zeros of f(x)=12x³+8x²-3x-2

Finding the Zeros of f(x)=3x5-10x4-6x³+24x²+11x-6 ;
Finding the Zeros of f(x)=3x³-x²+11x-20

Factoring Polynomial f(x)=x4-5x3-9x2+81x-108 using Rational Roots Theorem

Factoring a Polynomial f(x)=x5-5x4-5x3+45x2-108 using the TI-89 Calculator

Ch 3.5 Rational Functions

Graphing y=1/x-2 and f(x)= x²-4 / x-2 ;
As x→a¯ then f(x)→∞ ;
As x→a+ then  f(x)→∞;
As x→a¯ then f(x)→-∞ ;
As x→a+ then f(x)→-∞ ;
As x→-∞ then  f(x)→c;
As  x→-∞ then f(x)→c ;
As x→∞ then f(x)→c ;
As x→∞  thenf(x)→c ;

Steps of Graphing Rational Functions ;
Finding -intercept, Zeros, Vertical Asymptotes, Horizontal Asymptotes, Domain, and Graphing  f(x)= x²-x / 9x³-9x²-22x+8(Part 1)

Finding -intercept, Zeros, Vertical Asymptotes, Horizontal Asymptotes, and Domain, and Graphing  f(x)= x²-x / 9x³-9x²-22x+8 (Part 2) ;
Definition of Oblique Asymptotes ;
Finding Domain, y-intercept, Zeros, Vertical Asymptotes, Horizontal Asymptotes, Oblique Asymptotes, and Graphing f(x)= 2x²-x-3 / x-2

Finding the Vertical Asymptotes for Rational Function g(x)= 1 / x²-4x+4

Finding the Horizontal Asymptotes of the following Rational Functions
h(x)=1 / x²-4x+4;

g(x)= 7x4 / 5x4+9

Find the Intercepts of the Rational Function g(x)=x²+6x / 3x³-17x²-6x

Finding the Oblique Asymptote of the Rational Function f(x)= 4x³-5 / 2x²-6

Graphing the following Rational Functions g(x)= x-1 / x²+4x+4

f(x)= x²+x-42 / x²-x-30

Ch 3.6 Variation

Definition of Direct Variation ;
Real life of Direct Variation ;
y Varies Directly with x. y=54 when x=9. Determine the Direct Variation Equation and then Determine y when x=3.5;
Word Problem of Hooke’s Law of Spring Displacement ;
y Varies Directly with the Square of x.  y=32 when x=4. Determine the Direct Variation Equation and then Determine y when x=6

Definition of Inverse Variation ;
Real Life of Inverse Variation ;
y Varies Inversely with x.  y=4 when x=2. Determine the Inverse Variation Equation and then Determine y when x=16;
Word Problem of Emptying a Tank with a Pump ;
Word Problem of the Force Needed to Break a Board ;
y Varies Inversely as the Square Root of x. y=6 when x=16. Determine the Inverse Variation Equation and then Determine  y when x=4