Transcript 6.2 Polynomials and Linear Factors
Do Now
Factor completely and solve.
1. x 2 - 15x + 50 = 0 2. x 2 + 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
Relative Minimum
Polynomials and Real Roots
Relative Maximum ROOTS !
• POLYNIOMIAL EQUIVALENTS 1. Roots 2. Zeros 3. Solutions 4. X-Intercepts 5. Relative Maximum 6. 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 x 2 + 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 (x 2 +5x+6)(x+1)=x (x 2 +5x+6)+1 (x 2 +5x+6) = x 3 +6x 2 +11x+6 Standard form
We can also use the GCF (greatest common factor) to factor a poly in standard form into its linear factors.
Ex. 2x 3 +10x 2 +12x GCF is 2x so factor it out. We get 2x(x 2 +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 .
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
zero (solution) of the function. is a
and
If
a
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
zero (solution) of the function. is a 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 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
• • • • • •
Write a polynomial given the roots 2, -4, ½
Put in factored form y = (x – 2)(x + 4)(2x – 1) Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (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
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)=x 4 +6x 3 +8x 2 state the multiplicity and
Let’s Try One
• Find any multiple zeros of f(x)=x 4 +6x 3 +8x 2 state the multiplicity and
Equivalent Statements about Polynomials
-4 is a solution of x 2 +3x-4=0
-4 is an x-intercept of the graph of y=x 2 +3x-4
-4 is a zero of y=x 2 +3x-4
(x+4) is a factor of x 2 +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)(x 2 - 6x + 9) now distribute to get = x 3 - 4x 2 - 3x + 18 this function has zeros at -2,3 and 3
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)(
x
2 + 3
x
) =
x
(
x
2 + 3
x
) – 2(
x
2 + 3
x
) =
x
3 + 3
x
2 – 2
x
2 – 6
x
=
x
3 +
x
2 – 6
x
Multiply (
x
Multiply.
Simplify.
+ 3)(
x
).
Distributive Property The function ƒ(
x
) =
x
3 +
x
2 – 6
x
has zeros at 2, –3, and 0.
Polynomials and Linear Factors
Find any multiple zeros of ƒ(
x
) =
x
5 – 6
x
4 + 9
x
3 and state the multiplicity.
ƒ(
x
) =
x
5 – 6
x
4 + 9
x
3 ƒ(
x
) =
x
3 (
x
2 – 6
x
+ 9) Factor out the GCF,
x
3 .
ƒ(
x
) =
x
3 (
x
– 3)(
x
– 3) Factor
x
2 – 6
x
+ 9.
( Since you can rewrite
x
3
x
as (
x
– 0)(
x
– 0)(
x
– 0), or – 0) 3 , the number 0 is a multiple zero of the function, with multiplicity 3. Since you can rewrite (
x
– 3)(
x
– 3) as ( zero of the function with multiplicity 2.
x
– 3) 2 , the number 3 is a multiple
Assignment #7
pg 293 7-37 odds
Finding local Maximums and Minimum
• • • • • Find the local maximum and minimum of x 3 3x 2 – 24x + Enter equation into calculator Hit 2 nd Trace Choose max or min Choose a left and right bound and tell calculator to guess