1. Do Now Factor completely and solve. x2 - 15x + 50 = 0

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

3. Polynomials and Real Roots
Relative Maximum
POLYNIOMIAL EQUIVALENTS
Roots
Zeros
Solutions
X-Intercepts
Relative Maximum
Relative Minimum
Relative Minimum
ROOTS !

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

5. We can also take a polynomial in factored form and rewrite it into standard form. 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

6. We can also 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

7. The greatest y value of the points in a region is called the local maximum. The least y value among nearby points is called the local minimum.

8. 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,

9. Zeros A zero is a (solution or x-intercept) to a polynomial function.
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.

10. A repeated zero is called a multiple zero
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. A triple zero “flattens out” as it crosses the x axis.

11. 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

12. 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

13. 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

14. 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

15. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

16. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

17. 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

18. Example 1
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

19. 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.

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.

21. Assignment #7
pg odds

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