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U4 Day 1: Notes Polynomials, Linear Factors, and Zeros
Polynomial functions are _________________ ( )
and have _____________ and ______________ ______________
Ex. 1 Write in factored form Ex. 2 Write a polynomial function in standard and find the zeros. (7- 11 ) form with the given zeros. (19-25) 2 x^3 + 2 x^2 − 112 x = 0 3 ,0, 2 5
(^) x x = −
Review: End Behavior: x → −, f ( ) x →? x → +, f ( ) x →? (left) (up/down) (right) (up/down)
f ( ) x = positive xeven f ( ) x = negative xeven f ( ) x = positive xodd f ( ) x = negative xodd
f ( ) x = 5 x^8 ... f ( ) x = − 3 x^10 ... f ( ) x = 4 x^9 ... f ( ) x = − x^5 ...
x → −, f ( ) x → x → −, f ( ) x → x → −, f ( ) x → x → −, f ( ) x →
x → +, f ( ) x → x → +, f ( ) x → x → +, f ( ) x → x → +, f ( ) x →
Ex. 3 Find the zeros. Graph using End behavior, multiplicity, and max # of turn (#’s 13- 17 odd, 27 – 33 odd) a) f ( ) x = x^4 − 2 x^3^ − 8 x^2 …factor… f ( ) x =
Zeros : __________ ___________ ____________ (x-intercepts, solutions, roots)
End behavior: x → −, f ( ) x → ________ x → +, f ( ) x → __________
Multiplicity : When the same zero happens “multiple” times.
Tangency: ________ number of the same zeros…it ________________on the x-axis
Inflection: _________ number of the same zeros…it __________________the x-axis
Tangency (bounce): ___________ Inflection (cut through): _______________
Degree: ________________ exponent degree=______
Max number of turns : _____________________________ Max # of turns: ____
*discuss Relative Max & Relative Min (Local Max and Local Min)
Ex 5 f ( ) x = − 5 ( x − 1 ) ( x + 4 )^3 f ( ) x = ( ) ( ) ( ) ( )
Zeros : __________ ___________ (x-intercepts, solutions, roots)
End behavior: x → −, f ( ) x → ________ x → +, f ( ) x → __________
Multiplicity : When the same zero happens “multiple” times.
Tangency (bounce): ___________ Inflection (cut through): _______________
Max number of turns : degree - 1 (highest exponent - 1) Max # of turns: ____
*discuss Relative Max & Relative min (local max and local min)
Ex. 6 : Find the Relative Max or Relative Min to the nearest tenth on your calculator. (35-37)
a) f ( ) x = x^3 − 9 x b) f ( ) x = 3 x^3 + x^2 − 5 x
Relative max: __________ Relative Max:____________
Relative Min: __________ Relative Min: ____________
Unit 4 Day 2 Notes: Solving Polynomial Equations Name_______________________
The _______________ tells you the amount of ___________________ (real/imaginary) for the equation
x^3 − 5 x^2 + 7 x − 3 = 0 ____ solutions 4 x^2 + 8 x − x^4 + 17 = 0 _____ solutions
Ex. 1 Find the real or imaginary solutions of each equation by factoring.
a. 6 x^2 − 11 x = 7 b. x^3^ − 2 x^2 + 8 x − 16 = 0
c. 5 x^3 − 135 = 0 d. x^4 + 3 x^2 = 28
Ex 3 The product of three consecutive integers is 720. What are the numbers? SHOW ALL WORK
Variables: Equation:
First number = __________
Second number= _________
Third number = __________
Sentence: ________________________________________________________________________________
Ex 4 Use your calculator...Graph each function to find the zeros. Rewrite the function in factored form.
f ( ) x = 2 x^4 − 11 x^3 − 29 x^2 + 158 x − 120
Zeros: ________________________________________
Factors: _____________________________________
Function (factored form):
Unit 4 Day 3: Notes: Dividing Polynomials Name __________________________
Remember Long Division…
Quotient
Divisor (^) 21 674 dividend
Use Long division to divide a polynomial by another polynomial. Always make sure the _______________ and
________________________ are arranged in descending order. Leave a ______________ if you are missing a term.
Ex. 1 a. divide 2 a − 5 a^2 + 4 a^4 − 10 by 2a-3 or
a a a a
Answer: ______________________________________________________________________
Remainder Theorem: provides a quick way to find the remainder of a polynomial long-division problem.
If a polynomial, f(x), is divided by (x – k), then the remainder is f(k).
Ex. 2 Find the remainder without going through long division. Show your work.
a. (4 x^4^ + 3 x^2 − 2 x + 1) ( x −2) b. (^ )
x x x
Remainder = ___________ Remainder = _______________
Factor Theorem: helps you use the remainder to decide if you have a factor.
A polynomial, f(x), has a factor, (x – k), if and only if f(k) = 0.
Ex. 3 Determine if the divisor is a factor.
a. (^) ( 4 x^2 + 23 x − (^16) ) (^) ( x + (^5) ) b.
( )
x x x
Remainder = ___________ Remainder = ________________
Is (x+5) a factor? _____________ Is (x-7) a factor? _______________ Name one ordered pair: __________ Name one ordered pair: __________ Therefore, ___________ is a __________________
So…Long Division can help us find _________________________ of a polynomial.
Since it can help us find factors of a polynomial, it can also help us find _____________________________,
__________________________, _________________________________, and _________________________
Synthetic Division: Makes long division quicker …but only works for linear factors _________________. Once again…
Always make sure the _______________ and ________________________ are arranged in descending order.
Leave a ______________ if you are missing a term. You will only need the _____________________________
Ex. 4 Use Synthetic division to divide.
a.
( )
x x x x
b. ( x^5 − 2 x^3 + 2 − x^2 ) (^) ( x + (^3) )
____ ____ ____ ____ ____
_____________________________________ ______________________________________
These are the coefficients of the answer.
Answer: ______________________________________ Answer: ______________________________________
Is (x-4) a factor? _____________ Is (x+3) a factor? ____________
Is x = 4 a solution? ___________ Is x = - 3 a solution? __________
Ex. 5 Use the Synthetic Division and the Factor Theorem to completely factor the problem.
y = x^3 − 28 x − 48; ( x + 4 ) is one of the factors
How do you find the zeros and factors if they don’t start you with some of them (zeros/factors)? Guess and Check?
Rational Root Theorem : minimizes the amount to Guess and checks possibilities.
Factors of the Constant Term Factors of the Leading Coefficient
P
Q
…this only lists “possible” Rational roots
Ex. 4 Use the Rational Root Theorem to list all possible rational roots. Then find any actual rational roots.
f ( ) x = x^3^ − 8 x^2 + 17 x − 6
Rational Root Theorem: Factors of the Constant Term Factors of the Leading Coefficient
P
Q
= ____ = ___________________
“POSSIBLE” roots : __________________________________________________
**As a class, use the remainder and factor theorem to decide which is a root. What root is there? ____________
Now use synthetic division to divide out the root.
All roots/zeros: ______________________________________________________________________
Rational roots/zeros: ________________ Irrational roots/zeros_________________________________
Real roots/zeros: _________________________________________________________________
imaginary Roots/zeros: __________________________________________________________________
Ex. 5 Use the Rational Root Theorem to list all possible rational roots. Then find any actual rational roots.
P x ( ) = x^4 − x^3^ + 2 x^2 − 4 x − 8
Rational Root Theorem: Factors of the Constant Term Factors of the Leading Coefficient
P
Q
= ___ = _______________________
“POSSIBLE” roots : __________________________________________________
**As a class, use the remainder and factor theorem to decide which is a root. What root is there? _____________
Now use synthetic division to divide out the root.
All roots/zeros: ______________________________________________________________________
Rational roots/zeros: _______________________ Irrational roots/zeros:_________________________
Real roots/zeros: _________________________________________________________________
imaginary Roots/zeros: __________________________________________________________________
b. −2, 3 − 5 i …______________________ roots/zeros always come in ___________________ ____________
Function: _______________________________________________________________
c. 6, 2 − 7 …_________________________ roots/zeros always come in ___________________ __________
Function: _______________________________________________________________
Ex. 8 Is it possible to have the following situations? How do you know?
a. 1 real root, 2 complex roots and 3 irrational roots?
b. 4 real roots, 1 imaginary, and 4 rational roots?
c. roots that equal: −9, 0, 5, 8 − 3, 7i?
Unit 4 Day 5 Notes: The Fundamental Theorem of Algebra Name __________________________
Fundamental Theorem of Algebra:
Every _______________________ with a ____________________________ has exactly _____ roots in the
______________________ _______________________ system.
How many roots/zeros/solutions does the polynomial have?
a. f ( ) x = − 3 x + 4 b. P x ( ) = − 5 x^2 + 6 x − 4 c. y = − x^3 + 7 x − 9 d. f ( ) x = 8 x^7 − x + 2
______ roots ______ roots ______ roots ______ roots
Ex. 1 Find All the zeros/roots of the polynomial.
x^4 + 2 x^3^ = 13 x^2 − 10 x POSSIBLE Rational Roots: _____________________________
- Today, we will use our Calculator to help see the rational roots.
Which roots/zeros can you see? _______________________
**Use one of these roots/zeros to start your Synthetic Division. Even though you can see the zeros on the calculator, you must show how to get them using Synthetic Division.
All the Zeros/Roots/Solutions: _________________________________________________________
*Extra…. Factors: ___________________________________________________________________________
Ex. 2 Find All the zeros/roots of the polynomial.
p x ( ) = 7 x^2 + x^5^ − x^4^ − 7 x^3 − 18 x + 18 POSSIBLE Rational Roots: _____________________________
- Use your Calculator to help see the rational roots. Which roots/zeros can you see? _______________________
**Use one of these roots/zeros to start your Synthetic Division. Even though you can see the zeros on the calculator, you must show how to get them using Synthetic Division.
All the Zeros/Roots/Solutions: _________________________________________________________
*Extra… Factors: _________________________________________________________________
