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
Relative Maximum
Relative
Minimum
ROOTS !
•
1.
2.
3.
4.
5.
6.
POLYNIOMIAL
EQUIVALENTS
Roots
Zeros
Solutions
X-Intercepts
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+6
Standard 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.
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,
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 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
Note that the ½ term
•
•
•
•
•
•
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
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)
Ex: y =x(x-1)(x+3)
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
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
Polynomials and Linear Factors
Write a polynomial in standard form with
zeros at 2, –3, and 0.
2
–3
0
ƒ(x) = (x – 2)(x + 3)(x)
Zeros
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.
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.
Assignment #7
pg 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