Algebra 2, with Trig
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Chapter 3
Polynomials
Sections Covered:
3.1 Remainder and Factor Theorems 3.2 Analyzing Polynomial Graphs 3.3 Zeros of Polynomials 3.4 Fundamental theorem of Algebra 3.5 Graphs of Rational Functions ~ November 2013 ~ Sun Mon Tue Wed Thu Fri Sat 14 B 15 A 16 3.1 “The Remainder 3.1 “The Remainder and Factor Theorems” and Factor Theorems” Long Division Long Division
HW 3.1 Long Division HW 3.1 Long Division
17 18 B 19 A 20 B 21 A 22 B 23 3.1 “The Remainder 3.1 “The Remainder Quiz: Long Division Quiz: Long Division 3.3 “Zeros of and Factor Theorems” and Factor Theorems” 3.2 “Analyzing Graphs 3.2 “Analyzing Graphs Polynomial Functions” Synthetic Division Synthetic Division of Polynomials” of Polynomials” Rational Zeroes
HW 3.1 Synthetic HW 3.1 Synthetic Division Division HW 3.2 HW 3.2 HW 3.3 Rational Zeros 24 25 A 26 B 27 Thanksgiving 28 Thanksgiving 29 Thanksgiving 30 3.3 “Zeros of 3.3 “Zeros of Break Break Break Polynomial Functions” Polynomial Functions” Rational Zeroes Descarte’s Rule
HW 3.3 Rational HW 3.3 Descarte’s Zeros Rule ~ December 2013 ~ 1 2 A 3 B 4 A 5 B 6 A 7 3.3 “Zeros of Quiz 3.1-3.3 Review Quiz 3.1-3.3 Review ***Quiz: 3.1-3.3*** ***Quiz: 3.1-3.3*** Polynomial Functions” Descarte’s Rule
HW 3.3 Descarte’s HW Review 3.1-3.3 HW Review 3.1-3.3 Pre-Lab Worksheet Pre-Lab Worksheet Rule 8 9 B 10 A 11 B 12 A 13 B 14 3.4 “The Fundamental 3.4 “The Fundamental Bungee Barbie LAB Bungee Barbie LAB 3.5 “Graphs of Rational Theorem of Algebra” Theorem of Algebra” Functions”
HW 3.4 HW 3.4 HW 3.5
15 16 A 17 B 18 A 19 B 20 A 21 3.5 “Graphs of Rational Chapter 3 Test Review Chapter 3 Test Review ***Chapter 3 Test*** ***Chapter 3 Test*** Functions”
HW Chapter 3 Test HW Chapter 3 Test HW 3.5 Review Packet Review Packet Table of Contents Notes 3.1: Long Division pg. 3-4 Notes 3.3: Descartes Rule pg. 22-24
HW 3.1: Long Division pg. 5-6 HW 3.3: Descartes Rule pg. 25-26
Notes 3.1: Synthetic Division pg. 7-9 Notes 3.4: Fund. Thm of Algebra pg. 27-28
HW 3.1: Synthetic Division pg. 10-11 HW 3.4: Fund. Thm of Algebra pg. 29-30
Notes 3.2: Analyzing Graphs pg. 12-15 Notes 3.5: Graphs of Rat. Fun. pg. 31-33
HW 3.2: Analyzing Graphs pg. 16-17 HW 3.5: Graphs of Rat. Fun. pg. 34-36
Notes 3.3: Rational Roots pg. 18-19 HW Reflection Sheet pg. 37-38
HW 3.3: Rational Roots pg. 20-21
3.1 Long Division
We’ve learned how to multiply polynomials. Now, we’ll learn how to DIVIDE polynomials.
There are 2 ways to divide: long division and synthetic division. First, let’s go back to grade school and review how to divide plain old regular numbers.
8 65180 Check:
Now, let’s apply the same process to polynomials. Check your solution by multiplying the divisor by the quotient and adding the remainder.
EX 1] x 2 x3 2x 2 6x 9
EX 2] 2x 3 2 x3 7x 2 17x 3 You Try:
1. (m2 – 7m – 11) ÷ (m – 8)
2. (a2 – 28) ÷ (a – 5)
3. (2x2 – 17x – 38) ÷ (2x + 3)
4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)
5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k) 3.1 Synthetic Division and the Remainder and Factor Theorems
Synthetic division works differently.
Divide x3 2x 2 6x 9 by x – 2.
1. First, the divisor must be in the form: (x – k). In our example, k = 2.
2. If a term is missing, you MUST use a zero as a place holder.
3. Write the leading coefficient. Then multiply diagonally and add vertically, multiply and add, etc. ***FILL IN BELOW****
4. The answer is interpreted as follows: Work backwards. The last number is the remainder. The next number back is the constant. The next number back is the x coefficient. The next number back is the x 2 coefficient. The next number back is the x 3 coefficient. And so on.
To divide x3 2x 2 6x 9 by x – 2.
2 1 2 -6 -9
x 2 x c remainder Do not write two signs!
EX 3] Divide x3 14x 8 by (x 4) using synthetic division. The Remainder Theorem: If a polynomial P(x) is divided by x c, then the remainder equals P(c) .
TRANSLATION: Do synthetic division, the remainder is your answer.
In the example at the top of the previous page, divide x3 2x 2 6x 9 by x – 2, 5 we got x 2 + 4x + 2 + . x 2
The Remainder Theorem states that f (2) will be equal to –5!
Example:
The Factor Theorem: A polynomial function P(x) has a factor of x c if and only if P(c) 0 . That is, x c is a factor if and only if c is a zero of P.
Solutions = Roots = Zeros = x-Intercepts When we find the roots of a polynomial equation, we are finding the places where the value of the function is ZERO. If f (x) = 0, then x is a root!!!!
Examples: If f(4) = 0 then x = 4 is a zero and (x – 4) is a factor. If f(-7) = 0 then x = -7 is a zero and (x + 7) is a factor. If f(3/2) = 0 then x = 3/2 is a zero and (2x – 3) is a factor.
Watch what happens when we find f (3) for the function f (x) = x2 + 2x − 15
3 1 2 −15 Is your remainder 0? ______That means ______is a root of the equation, a zero of the function, and an x-intercept on the graph!! That also means that ______is a factor of the polynomial. And it means that ______is also a factor, also know as a reduced polynomial or a depressed polynomial. Getting zero in synthetic substitution is a big deal!! EX 4] f (x) = x3 + x2 + 2x + 24. *******(NEED CALCULATOR)*******
Graph the function. (You will need to set your window.) There is one real root at x 3 . Because the degree of the polynomial is 3, we know there are two other roots. They must be imaginary.
We will use synthetic substitution to divide out (x + 3). Then the quadratic formula will allow us to find the other roots.
−3 1 1 2 24 So, using the coefficients of the quotient, we write ______(x + 3)(x2 − 2x + 8) = 0
We already know x 3 . We use the quadratic formula on the second ( ) and get
x = = = b b 2 4ac x 2a = = 1 i 7 .
Therefore, x 3 , 1 i 7 , 1 i 7 Do you remember “complex conjugates”?
Example Problems:
Use Synthetic Division: 1. (m2 – 7m – 11) ÷ (m – 8) 3. (2x2 – 17x – 38) ÷ (2x + 3) 5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k)
2. (a2 – 28) ÷ (a – 5) 4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)
Use the Remainder Theorem to find P(c).
6. P(x) = 2x3 – x2 + 3x – 1 , c = 3 7. P(x) = 6x3 – x2 + 4x , c = -3 8. P(x) = -x3 + 3x2 + 5x + 30 , c = 8
Determine if 3.2 Analyzing Graphs of Polynomials
3.3 Possible Rational Zeros NOTE: P(x) is a polynomial function. ALL polynomial functions have graphs that are ______.
Zeros of function P(x) : x – values for which p(x) = 0
Roots of equation P(x) : x – values for which p(x) = 0
Multiple Zeros of a Polynomial Function:
If P(x) has (x r) as a factor exactly k times, then r is a zero of multiplicity k of P(x) .
(x- r )k muliplicity
Zero/Root
EX 1] Find the zeros of P(x) and state the multiplicity of each zero. P(x) x 7 2 x 8 4 x 5 3 2x 1
_____ occurs as a zero of multiplicity _____ . _____ occurs as a zero of multiplicity _____ . _____ occurs as a zero of multiplicity _____ . _____ occurs as a zero of multiplicity _____ . y f (x) Even Root = Bounce Point x
Odd Root = goes through
EX 2] Given P(x) x 7 2 x 8 4 x 5 3 2x 1.
a) Find the maximum number of roots. ______
b) Which zero(s) create a bounce point? ______
c) Which zero(s) does the graph go through? ______The Rational Zero Theorem:
WHY IS IT IMPORTANT: Narrows the search for rational zeros to a finite list.
n n1 If P(x) an x an1 x ⋯ a1 x a0 has integer coefficients an 0 p and is a rational zero (in lowest terms) of p, then q
p is a factor of the constant term a0 and q is a factor of the leading coefficient an .
EX 3] Find the roots of x3 6x 2 10x 3 0.
HINT: Apply the Rational Root Theorem to find the possible rational roots!
What is p? ______
What is q? ______
HINT: Use synthetic division on x3 6x 2 10x 3 0 to locate a root! (Use your calc to estimate a zero )
EX 4] Find the possible rational roots of 3x3+ 5 x 2 + 7 x + 2 = 0 .
HINT: Apply the Rational Root Theorem to find the possible rational roots!
What is p? ______
What is q? ______3.3 Descartes’ Rule of Signs
Descartes’ Rule of Signs:
WHY IS IT IMPORTANT: Narrows down even further the possible positive and negative roots.
Let P(x) be a polynomial function with real coefficients and with terms arranged in order of decreasing powers of x.
The number of positive real zeros of P(x) is equal to the number of variations in the sign of P(x), or to that number decreased by an even integer. The number of negative real zeros of P(x) is equal to the number of variations in sign of P(-x), or to that number decreased by an even integer.
Ex 5] Use Descartes’ Rule of signs to determine both the number of possible positive and the number of possible negative real zeros of each polynomial function.
a) P( x )= x4 - 5 x 3 + 5 x 2 + 5 x - 6 b) P( x )= 2 x5 + 3 x 3 + 5 x 2 + 8 x + 7
Number of variations: ______Number of variations: ______
Number of possible positive real zeros: ______Number of possible positive real zeros: ______
P(-x) = P(-x) =
Number of variations: ______Number of variations: ______
Number of possible negative real zeros: ______Number of possible negative real zeros: ______Steps for Finding the Zeros of a Polynomial Function with Integer Coefficients: 1) Gather General Information. Determine the degree n of the polynomial function. The number of distinct zeros of the polynomial function is at MOST n. Apply Descartes’ Rule of Signs to determine the number of possible negative real zeros of each polynomial. 2) Check rational zeros. Apply the Rational Zero Theorem to list rational numbers that are possible zeros. Use synthetic division to test the numbers in the list. 3) Work with the reduced/depressed polynomial. Each time a zero is found, obtain the reduced/depressed polynomial. Work to get a reduced polynomial of degree 2. Then, find its zeros by factoring or by applying the quadratic formula. Quick Sketch EX 6] Find the zeros of f (x) x3 7x 2 16x 12 .
At most ______zeros.
Rational Root Theorem – Possible rational zeros:
Descartes Rule:
Synthetic Division/Quad. Formula/Factoring y
x 5 EX 7] Find the zeros of g( x )= 3 x4 + 23 x 3 + 56 x 2 + 52 x + 16 Quick Sketch
y
x 5 3.4 The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra: If P(x) is a polynomial function of degree n 1 with complex coefficients, then P(x) has at least one complex zero.
The Linear Factor Theorem:
If P(x) is a polynomial function of degree n 1 with leading coefficient an 0 , then P(x) has exactly n linear factors.
P(x) an x c1 x c2 x c3 ⋯x cn where c1, c2, … , cn are complex numbers. (real and/or imaginary)
The Number of Zeros of a Polynomial Function Theorem: If P(x) is a polynomial function of degree n 1, then P(x) has exactly n complex zeros, provided each zero is counted according to its multiplicity.
EX 1] Find all zeros of the polynomial function and write the polynomial as a product of linear factors. P(x) x 4 6x3 10x 2 2x 15
The Conjugate Pair Theorem: If a bi (b 0 ) is a complex zero of a polynomial function with real coefficients, then the conjugate a bi is also a complex zero of the polynomial function.
Example: If (5i - 3) is a zero, then ______is also a zero. If 8i is a zero, then ______is also a zero. EX 2] Find a polynomial function P(x) that has the indicated zeros.
a) degree 3; 1, 2, and -3 as zeros
b) degree 3; real coefficients and zeros 2i and -3
c) degree 2; real coefficients and a zero is 3 – 7i
Practice: Find all the zeros for the equation x 4 5x 3 4x 2 3x 9 0 . -1 +i 3 - 1 - i 3 Hint: The two complex zeros are: x = , 2 2 3.5 Graphs of Rational Functions
p (x) f (x) p(x) q(x) Rational Function can be written in the form q (x) where and are polynomials and q(x) is NOT the zero polynomial.
Note The domain of a rational function of x includes all real numbers except the x-values that would make the denominator equal to zero.
The Graph of a Rational Function: 1. Removable Discontinuity (Hole in the graph) occurs when p(x) and q(x) have a common factor
2. Non-removable Discontinuity (Vertical Asymptote) occurs when the denominator equals zero
3. Horizontal Asymptote the value that the function approaches as x increases without bound
a. If the degree of the numerator < the degree of the denominator; y = 0 (the x-axis) is the horizontal asymptote
b. If the degree of the numerator = the degree of the denominator; lead coefficient of the numerator y is the H.A. lead coefficient of the denominator
c. If the degree of the numerator > the degree of the denominator; there is NO horizontal asymptote
4. x-intercept zero(s) of the numerator
5. y-intercept the value of f(0) y 6. Slant Asymptote occurs when the degree of the numerator is EXACTLY one more than the degree of its denominator x 2 x ex: f (x) x 1 **Use long division to find the equation of the slant asymptote. x The slant asymptote will always be linear! Do NOT include the remainder x 2 9 1. Graph f (x) Advanced Functions - 3.5 Notes x 3 Hole in graph: ______
V.A. ______
H.A. ______
x-intercept ______
y-intercept ______
slant asymptote ______
domain ______range ______
y 3x 2 17x 20 2. Graph g(x) x 2 5x 4
Hole in graph: ______x
V.A. ______
H.A. ______
x-intercept ______y y-intercept ______
slant asymptote ______
domain ______range ______x 3x 18 f (x) 3. Graph x3 4x 2 12x Hole in graph: ______
V.A. ______
H.A. ______
x-intercept ______
y-intercept ______
slant asymptote ______
domain ______range ______
2x 2 x 4. Graph g(x) x 1 y
Hole in graph: ______
V.A. ______x
H.A. ______
x-intercept ______
y-intercept ______
slant asymptote ______
domain ______range ______Chap. 3 Homework Completion Sheet Name: ______Block: ______
Prerequisite Skills: 1. Long Division 2. Factoring 3. Using Complex Numbers 4. Solving Using Quadratic Formula
Notes Date Assignment Score What am I confident about? (out of 20) Specifically, what was difficult or confusing? Am I weak on a prerequisite skill?