1. Do Now Given the polynomial 8 -17x3+ 16x - 20x4 - 8
Write the polynomial in standard form: __________
Classify the polynomial by degree: _______________________________________
Classify the polynomial by the number of terms: ________________________________
State the End Behavior: ________________________________
- 20x4 -17x3 +16x
Quartic
Trinomial
Down and Down

2. 5.2 Polynomials, Linear Factors, and Zeros
Learning Target: I can find the zeros of a polynomial function
and
I can write the function from its zeros
Learning Target: I can find the zeros of a polynomial function
and
I can write the function from its zeros

3. Polynomials and Real Roots
Relative Maximum
Let’s Review
Roots
Zeros
Solutions
X-Intercepts
Relative Maximum
Relative Minimum
Relative Minimum
ROOTS !
LT: I can find the zeros of a polynomial function
and
I can write the function from its zeros L.

4. Linear Factors
Just as you can write a number into its prime factors (jail trick: Example 6 into 2 & 3) you can write a polynomial into its linear factors. (factoring) x2 + 4x – 12 into (x+6)(x-2)

5.  Important Linear Factors
Take some private time and try factoring the following so we can find the zeros: 2x3 + 8x2 – 24x = 0 Factor out the GCF 2x(x2 + 4x – 12) = 0 Factor the Quadratic 2x(x-6)(x+2) = 0 Use the Zero Product Property x = 0, 6 or -2
 Important

6. We can also take a polynomial in factored form and rewrite it into standard form. Take a minute on your own and try to distribute Ex. (x+1)(x+2)(x+3) = foil distribute (x2+5x+6)(x+1)=x (x2+5x+6)+1 (x2+5x+6) = x3+6x2+11x+6 Standard form
 Book way
 My way x2 5x 6 X x3
5x2 6x 1

7. Vocabulary Alert. A repeated zero is called a multiple zero
Vocabulary Alert! A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs. On a graph, a double zero “bounces” off the x axis. (the parent quadratic) A triple zero “flattens out” as it crosses the x axis. (the parent cubic)

8. Zeros A zero is a (solution or x-intercept) to a polynomial function.
Zero Product Property
If (x – a) is a factor of a polynomial, then a is a zero (solution) of the function.
If a polynomial has a repeated solution, it has a multiple zero.
The number of repeats of a zero is called its multiplicity.

9. Zeros Copy down the equation: (x+1)(x-7)(x-2) What are the zero’s?
If we use the zero product property we get:
X= -1, 7 or 2

10. Vocab!! What is Multiplicity?
Multiplicity is when you have multiple roots that are exactly the same. We say that the multiplicity is how many duplicate roots that exist.
Ex: (x-2)(x-2)(x+3)
Ex: (x-1)4 (x+3)
Ex: y =x(x-1)(x+3)
Note: two answers are x=2; therefore the multiplicity is 2
Note: four answers are x=1; therefore the multiplicity is 4
Note: there are no repeat roots, so we say that there is no multiplicity

11. Polynomials and Linear Factors
Write a polynomial in standard form with zeros at 2, –3, and 0.
2 –3 0 Zeros
ƒ(x) = (x – 2)(x + 3)(x) Write a linear factor for each zero.
= (x – 2)(x2 + 3x) Multiply (x + 3)(x).
= x(x2 + 3x) – 2(x2 + 3x) Distributive Property
= x3 + 3x2 – 2x2 – 6x Multiply.
= x3 + x2 – 6x Simplify.
The function ƒ(x) = x3 + x2 – 6x has zeros at 2, –3, and 0.
LT: I can find the zeros of a polynomial function
and
I can write the function from its zeros

12. Write a polynomial given the roots 0, -3, 3
Put in factored form
y = (x – 0)(x + 3)(x – 3)
y = (x)(x + 3)(x – 3)
y = x(x² – 9)
y = x³ – 9x
L.T. I can analyze the factored form of a polynomial and write function from its zeros

13. Graph / Sketch Y = (x-2)(x+9)(x+2) We know there are zeros at?
Can you describe the end behavior?
How many turns are there?
Sketch it!

14. Assignment #7 Friday pg 293 9-33 by 3’s Do not Graph Monday
Graph when instructed

15. Write a polynomial given the roots 2, -4, ½
Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (2x-1)
Put in factored form
y = (x – 2)(x + 4)(2x – 1)
y = (x² + 4x – 2x – 8)(2x – 1)
y = (x² + 2x – 8)(2x – 1)
y = 2x³ – x² + 4x² – 2x – 16x + 8
y = 2x³ + 3x² – 18x + 8
L.T. I can analyze the factored form of a polynomial and write function from its zeros

16. Write the polynomial in factored form. Then find the roots
Write the polynomial in factored form. Then find the roots. Y = 3x³ – 27x² + 24x
Y = 3x³ – 27x² + 24x
Y = 3x(x² – 9x + 8)
Y = 3x(x – 8)(x – 1)
ROOTS?
3x(x – 8)(x – 1) = 0
Roots = 0, 8, 1
FACTORED FORM
L.T. I can analyze the factored form of a polynomial and write function from its zeros

17. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity
L.T. I can analyze the factored form of a polynomial and write function from its zeros

18. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity
L.T. I can analyze the factored form of a polynomial and write function from its zeros

19. Equivalent Statements about Polynomials
-4 is a solution of x2+3x-4=0
-4 is an x-intercept of the graph of y=x2+3x-4
-4 is a zero of y=x2+3x-4
(x+4) is a factor of x2+3x-4
These all say the same thing
L.T. I can analyze the factored form of a polynomial and write function from its zeros

20. Polynomials and Linear Factors
Find any multiple zeros of ƒ(x) = x5 – 6x4 + 9x3 and state the multiplicity.
ƒ(x) = x5 – 6x4 + 9x3
ƒ(x) = x3(x2 – 6x + 9) Factor out the GCF, x3.
ƒ(x) = x3(x – 3)(x – 3) Factor x2 – 6x + 9.
Since you can rewrite x3 as (x – 0)(x – 0)(x – 0), or
(x – 0)3, the number 0 is a multiple zero of the function, with multiplicity 3.
Since you can rewrite (x – 3)(x – 3) as (x – 3)2, the number 3 is a multiple
zero of the function with multiplicity 2.
L.T. I can analyze the factored form of a polynomial and write function from its zeros

21. Example
We can rewrite a polynomial from its zeros. Write a poly with zeros -2, 3, and 3 f(x)= (x+2)(x-3)(x-3) foil = (x+2)(x2 - 6x + 9) now distribute to get = x3 - 4x2 - 3x + 18 this function has zeros at -2,3 and 3
L.T. I can analyze the factored form of a polynomial and write function from its zeros

22. Finding local Maximums and Minimum
Find the local maximum and minimum of x3 + 3x2 – 24x
Enter equation into calculator
Hit 2nd Trace
Choose max or min
Choose a left and right bound and tell calculator to guess
L.T. I can analyze the factored form of a polynomial and write function from its zeros

23. We can use the GCF (greatest common factor) to factor a poly in standard form into its linear factors. Ex. 2x3+10x2+12x GCF is 2x so factor it out. We get 2x(x2+5x+6) now factor once more to get 2x(x+2)(x+3) Linear Factors
L.T. I can analyze the factored form of a polynomial and write function from its zeros

24. Theorem
The expression (x - a) is a linear factor of a polynomial if and only if the value a is a zero (root) of the related polynomial function. If and only if = the theorem goes both ways If (x – a) is a factor of a polynomial, then a is a zero (solution) of the function. and Ifa is a zero (solution) of the function then (x – a) is a factor of a polynomial,
L.T. I can analyze the factored form of a polynomial and write function from its zeros