Pre-AP/GT Pre-Calculus Assignment Sheet
Unit 1.2 - Polynomials
September 2nd – 19nd, 2016
Date
Friday
9/2
Tuesday
9/6
Objective
Polynomial Long and Synthetic Division
Homework
Finish pages 2 – 3 and top of 4
Monday, September 5 –Labor Day – No School!
Polynomial Long and Synthetic Division and
Pages 6 – top of 7
Writing Polynomial Equations
Descartes Law of Signs and IVT
Quiz: Retention of Unit 1.1
Pages 10 – 11
Finish Descartes Law of Signs and IVT
Finish Pages 10 – 11
Rational Root Theorem (day 1)
Quiz: Dividing, Writing Polynomials, PNI Charts
Begin Page 14
Monday
9/12
Rational Root Theorem (day 2)
Finish Page 14
Tuesday
9/13
Graphing Polynomials using signs
Activity/WS
Polynomial Inequalities
Finish Page 15
Polynomial Inequalities
Quiz: Rational Root Theorem, Polynomial Inequalities
Finish Pages 16 – 17
Review Unit 1.2
Quiz: Homework
Study!
Wednesday
9/7
Thursday
9/8
Friday
9/9
Wednesday
9/14
Thursday
9/15
Friday
9/16
Monday
9/19
Unit 1.2 Test
1
Notes on Polynomial and Synthetic Division
Back in the Glory Days of Algebra II, you learned two types of Polynomial Divisions: Polynomial and Long.
Both are used heavily in calculus and have many applications. Let’s begin by reviewing Polynomial Long
Division.
Use Polynomial Long Division for the following:
1.
2.
3.
Use Synthetic Division for the following: Synthetic division is more commonly used when dividing by a
“binomial” because it is considered quicker and easier.
4. (2 x 3  3x 2  3x  2)  ( x  1)
6.
(3x 3  x 2  4)
x3
5. (2 x 3  7 x 2  8x  16)  ( x  4)
7.
3x 3  4 x 2  5
3
x
2
2
Rational Root Theorem: Just like normal division, if you divide two polynomials and the remainder is zero, it
means that you have found a factor (ex: if you divide 12 by 3 the remainder is zero. Therefore, 3 is a factor of
12.)
8. Determine which is a “factor” of
a. x – 2
9. Given that x + 2 is a factor of
. There could be more than one answer.
b. x – 3
c. 2x – 3
, find the remaining factors and zeros.
Find the value of “k” that makes the binomial a factor of the polynomial.
10. (x – 3), x 4  3x 3  kx2  x  15
11. (3x + 2), 15 x 4  10 x 3  6 x 2  kx  14
Factor Theorem: Says the value of f(x) at x = a is the same as the remainder when the polynomial is divided
by x – a.
12. Find f(2) if
(show two ways)
3
is divided by x – 1?
13. What is the remainder when
For Brownie Points  Find the value of “k” that makes the binomial a factor of the polynomial.
( x  2 ), x 3  3 x 2  kx  14
Notes on Polynomial and Synthetic Division
Degree of a polynomial and its meaning:
Applications of Polynomials:
Find the remaining factor(s) of the polynomials with the given factors: Write the complete factorization
of each polynomial and find all the real solutions.
1.
3.
; (x – 2), (x + 4)
2.
; (x – 1)
; (x + 2) (x – 4)
Add to your assignment on your own paper: Find the value of the following functions. Use synthetic
division.
1. f ( x)  2 x 3  4 x  2 , find f(-1).
2. f ( x)   x 4  3x 3  x 2  6 x  2 , find f(-3).
1
3. g ( x)   x 2  4 x  8 , find f(2).
4.
, find f(0).
2
4
Notes on Writing Polynomials
Writing the Equation of a polynomial.
The general form of a polynomial is y  a(x  r1 )(x  r2 )(x  r3 )...(x  rn ) , where “r” is the roots of the polynomial,
and “a” is the constant multiplier.
1. Write the equation of a polynomial with the given roots and goes through the given point:
x  3,  5, 2
(4, -1)
2. Write the polynomial equation that has roots 2 (triple root), 0, and -1 and passes through 1,1
Write the equation for the following graphs. Leave in factored form.
3.
4.
More Fun with Polynomials:
6. Sketch a graph of a negative odd function with five real
7. Given the polynomial and graph, find the roots:
roots that only crosses the x-axis three times.
p( x)  x 4  2 x3  8x 2  8x  16
5
Assignment on Writing Polynomials
Write the equation for the following graphs. Leave in factored form.
1.
________________________________________
3.
________________________________________
2.
____________________________________________
4.
____________________________________________
Write the equation of the polynomial with the given roots and point.
5. x = 3, -1, 4 (-4, 1)
6. Write the polynomial equation that has roots 5 (double root), -3, and passes through 3,1
6
7. Use synthetic division to find p (2) where p( x)  2 x5  3x3  12 x  3 .
8. Is  x  3 a factor of f ( x)  x 4  2 x3  3x  4 ? Support your answer with synthetic division and explain how you
determined your answers.
9. Find the value of “k” for
2 x2  5x  7
xk
where the remainder is 0.
Notes on Descartes Law of Signs and the Intermediate Value Theorem
Descartes Law of Signs:
Descartes Law of Signs is a method used to help eliminate possible roots and factors of polynomials. To use Descartes
Law of Signs, we create what is called a PNI Chart.
P-N-I Chart
A PNI Chart is used to help identify possible roots of a polynomial that are Positive, Negative, or Imaginary. To create a
PNI Chart, we follow these simple steps of a given polynomial of degree “n”.
1. There are as many positive real roots of P(x) as the number of sign changes of P(x) or less than
that by an even number.
2. There are as many negative real roots of P(x) as the number of sign changes of P(x) or less than
that by an even number.
3. To determine the number of imaginary roots, add the number of positive and negative roots in each
row of the PNI chart then subtract that sum from the degree of P(x) . When finished, each row
of the PNI chart must add up to the degree of P x .
7
Create a PNI Chart for the following:
1. P( x)  5 x3  8 x 2  4 x  3
2. P( x)  x5  6 x 4  3x3  7 x 2  8 x  1
3. P( x)  7 x6  4 x5  2 x3  8 x 2
Write a Polynomial with the given roots and goes through the given point.
1. x  3, x  3i, x  3i (0, 3)
2. x  1  2i, x  2(m3)
(1, 3)
8
Intermediate Value Theorem:
Ex: Given f ( x)  x 2  4 x  3 , how can we show that there is a root (x-intercept) between x = -2 and x = 0?
Use the Synthetic Division Chart below of the given function to answer the following questions.
1. What is the function that is being divided?
2. Is there a root(s)? Where?
3. What is the y-intercept?
4. Evaluate:
P(-3) =
P(3) =
5. How can we show that there is a root between x = -2 and x = -1?
9
Assignment on Descartes Law of Signs
For (1-5) Create a PNI Chart
1. P( x)  x 4  3x3  2 x 2  1
2. P( x)  4 x3  6 x 2  8 x  5
4. P( x)  x8  6 x7  3x6  7 x5  8 x 4  x3
3. P( x)  x5  x3  x  1
5. P( x)  x5  x3  x 2  x  2
Given the root(s), find the remaining roots
6. P( x)  x 4  3x 2  4 x 2  6 x  4  0 Roots: x = 1, 2
*7. x 6  2 x 5  4 x 4  2 x 3  5 x 2  0 Roots: x = 1, -1
Use the Intermediate Value Theorem for the following
8. Given: f ( x)  3x3  4 x  1 . Show that there is a root between x = -2 and x = -1.
9. Can the Intermediate Value Theorem be used the show that there is a root between x = 1 and x = 2 for the function:
f ( x)  x 2  2 x  2 . Explain why.
10
Use the Synthetic Division Chart to answer the following questions
10. What is the polynomial being divided?
11. PNI Chart:
12. Is there a root(s)? Where?
13. Can we show that there is a root between x = 4 and x = 3? Why or Why not, explain.
14. Can we show that there is a root between x = -3 and x = -4? Why or Why not, explain.
15. What is the y-intercept?
16. P(3) = ____
P(-1) = ____ P(-5) = ____
17. What is the remainder if the polynomial is divided by (x-1)?
18. Use a calculator to find the roots of f ( x)  2 x3 7 x  1 . Round answers to three decimal places.
11
Notes on Rational Root Theorem
The Rational Root Theorem: Given a polynomial function, then the rational roots of the polynomial are of the form

p
q
where p = a factor of the constant term and q = a factor of the leading coefficient.
1: f ( x)  x 4  x 3  x 2  3x  6
Possible rational roots:
Now use synthetic division to determine which roots are actual roots, then find remaining all roots.
2. f ( x)  2 x 3  5 x 2  12 x  5
What are the possible rational roots? Find all roots.
Factor the following completely: Show all work.
3.
f ( x)  x 5  3 x 4  4 x 3  4 x 2  3 x  1
4. f ( x)  x 3  9 x 2  20 x  12
12
Irrational and Imaginary Solutions:
Sketch the graph of f ( x)  x 2  16 . What do you notice? Does it cross the x – axis?
How to solve functions with Irrational and Imaginary Roots:
f ( x)  x 2  16
Find all of the zeros of the following: Show all work.
5. f ( x)  x 2  5 x  2
6. f ( x)  x 2  2 x  8
5. f ( x)  x 4  6 x 3  10 x 2  6 x  9
6. f ( x)  3x 3  4 x 2  8 x  8
Use the Graph below to factor the polynomial: f ( x)  4 x 4  24 x3  31x 2  30 x  45
13
Assignment on Rational Root Theorem
Factor and find all the Zero’s using the Rational Root Theorem.
1. f ( x)  x 3  6 x 2  11x  6
2. f ( x)  x 3  7 x  6
4. f ( x)  2 x 3  3x 2  1
5. f ( x)  9 x 4  9 x 3  58x 2  4 x  24
6. f ( x)  x 3  x 2  4 x  4
7. f ( x)  4 x 3  15 x 2  8 x  3
3. f ( x)  x 3  12 x 2  21x  10
Find the zeros of the following and write out the complete factorization as a product of linear factors.
8. f ( x)  x 2  25
9. f ( x)  x 2  4 x  1
10. f ( x)  x 4  81
11. f ( x)  x 2  2 x  2
12. f ( x)  x 3  6 x 2  13x  10
13. f ( x)  x 3  x  6
14. f ( x)  5x 3  9 x 2  28x  6
15. f ( x)  x 4  4 x 3  8 x 2  16 x  16
Find all the real solutions:
16. x 4  x 3  2 x  4  0
17. 2 x 4  7 x 3  26 x 2  23x  6  0
Use Descartes Law of Signs to determine the possible number(s) of positive and negative zeros.
18. f ( x)  5 x 3  x 2  x  5
Answer the following:
19. Use the Graph below to factor the polynomial: p( x)  x 4  2 x3  8x 2  8x  16
14
Notes of Applications of Polynomial Graphs
Determine the domains (use interval notation) for which the following functions are: (Show all work)
a. Above the x-axis
b. Below the x-axis
1.
2.
3.
4.
Write the polynomial with the given roots with the given degree:
5. x = 3, x = -1
7. x = 4, x = -3, x = 1
Degree = 2
6. x = -2
Degree = 2
degree = 5
15
Notes on Solving Polynomial Inequalities
What is a Critical Number?
Sketch the polynomials below, and use them to solve the inequalities:
1. ( x  3) 2 ( x  1)  0
2.  3x( x  2) 3 ( x  5) 2  0
Let’s be slick about it 
1. ( x  1) 2 ( x  3) 2 ( x  1)  0
3.  2( x  3) 2 ( x  4)  0
6. x 2  25
2.  2(2 x  1) 3 ( x  4)  0
4.
3 4
x ( x  3)( x  5)3  0
4
5. ( x 2  5 x)( x 2  16)  0
7. ( x  3) 2  16
16
Now some fun ones.
5.
x 2  3x  3  0
7. 3x 3  9 x 2  0
*6. x 2  3 x  5  0
8. x 3  3x 2  4 x  8  0
9.  x 3  2 x 2  4 x  5  3
17