Do Now

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

2.x2 + 10x – 24 = 0

5.2 Polynomials, Linear Factors, and Zeros

Learning Target: I can analyze the factored form of a polynomial and

write function from its zeros

Polynomials and Real Roots

• POLYNIOMIAL EQUIVALENTS

1. Roots2. Zeros3. Solutions4. X-Intercepts5. Relative Maximum6. Relative MinimumROOTS !

Relative Maximum

Relative Minimum

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)

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+6Standard form

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

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.

TheoremThe 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.

andIfa is a zero (solution) of the function then (x – a) is a

factor of a polynomial,

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.

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.

Write a polynomial given the roots0, -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

Write a polynomial given the roots2, -4, ½

• 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

Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (2x-1)

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

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)

Note: two answers are x=2; therefore the multiplicity is 2

Ex: (x-1)4 (x+3)

Note: four answers are x=1; therefore the multiplicity is 4

Ex: y =x(x-1)(x+3)

Note: there are no repeat roots, so we say that there is no multiplicity

Let’s Try One

• Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

Let’s Try One

• Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

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

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

Write a polynomial in standard form with zeros at 2, –3, and 0.

Polynomials and Linear Factors

= (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.

2 –3 0 Zeros

ƒ(x) = (x – 2)(x + 3)(x) Write a linear factor for each zero.

Find any multiple zeros of ƒ(x) = x5 – 6x4 + 9x3 and state

the multiplicity.

Polynomials and Linear Factors

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.

Assignment #7pg 293 7-37 odds

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